Analysis I

• 1.I $. 3 \mathrm{D} \quad$

  Analysis I

  Part IA, 2001

  What does it mean to say that $u_{n} \rightarrow l$ as $n \rightarrow \infty$ ?

  Show that, if $u_{n} \rightarrow l$ and $v_{n} \rightarrow k$, then $u_{n} v_{n} \rightarrow l k$ as $n \rightarrow \infty$.

  If further $u_{n} \neq 0$ for all $n$ and $l \neq 0$, show that $1 / u_{n} \rightarrow 1 / l$ as $n \rightarrow \infty$.

  Give an example to show that the non-vanishing of $u_{n}$ for all $n$ need not imply the non-vanishing of $l$.

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• 1.I.4D

  Analysis I

  Part IA, 2001

  Starting from the theorem that any continuous function on a closed and bounded interval attains a maximum value, prove Rolle's Theorem. Deduce the Mean Value Theorem.

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. If $f^{\prime}(t)>0$ for all $t$ show that $f$ is a strictly increasing function.

  Conversely, if $f$ is strictly increasing, is $f^{\prime}(t)>0$ for all $t$ ?

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• 1.II.9D

  Analysis I

  Part IA, 2001

  (i) If $a_{0}, a_{1}, \ldots$ are complex numbers show that if, for some $w \in \mathbb{C}, w \neq 0$, the set $\left\{\left|a_{n} w^{n}\right|: n \geq 0\right\}$ is bounded and $|z|<|w|$, then $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges absolutely. Use this result to define the radius of convergence of the power series $\sum_{n=0}^{\infty} a_{n} z^{n}$.

  (ii) If $\left|a_{n}\right|^{1 / n} \rightarrow R$ as $n \rightarrow \infty(0<R<\infty)$ show that $\sum_{n=0}^{\infty} a_{n} z^{n}$ has radius of convergence equal to $1 / R$.

  (iii) Give examples of power series with radii of convergence 1 such that (a) the series converges at all points of the circle of convergence, (b) diverges at all points of the circle of convergence, and (c) neither of these occurs.

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• 1.II.10D

  Analysis I

  Part IA, 2001

  Suppose that $f$ is a continuous real-valued function on $[a, b]$ with $f(a)<f(b)$. If $f(a)<v<f(b)$ show that there exists $c$ with $a<c<b$ and $f(c)=v$.

  Deduce that if $f$ is a continuous function from the closed bounded interval $[a, b]$ to itself, there exists at least one fixed point, i.e., a number $d$ belonging to $[a, b]$ with $f(d)=d$. Does this fixed point property remain true if $f$ is a continuous function defined (i) on the open interval $(a, b)$ and (ii) on $\mathbb{R}$ ? Justify your answers.

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• 1.II.11D

  Analysis I

  Part IA, 2001

  (i) Show that if $g: \mathbb{R} \rightarrow \mathbb{R}$ is twice continuously differentiable then, given $\epsilon>0$, we can find some constant $L$ and $\delta(\epsilon)>0$ such that

  $$\left|g(t)-g(\alpha)-g^{\prime}(\alpha)(t-\alpha)\right| \leq L|t-\alpha|^{2}$$

  for all $|t-\alpha|<\delta(\epsilon)$.

  (ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be twice continuously differentiable on $[a, b]$ (with one-sided derivatives at the end points), let $f^{\prime}$ and $f^{\prime \prime}$ be strictly positive functions and let $f(a)<0<f(b)$.

  If $F(t)=t-\left(f(t) / f^{\prime}(t)\right)$ and a sequence $\left\{x_{n}\right\}$ is defined by $b=x_{0}, x_{n}=$ $F\left(x_{n-1}\right) \quad(n>0)$, show that $x_{0}, x_{1}, x_{2}, \ldots$ is a decreasing sequence of points in $[a, b]$ and hence has limit $\alpha$. What is $f(\alpha)$ ? Using part (i) or otherwise estimate the rate of convergence of $x_{n}$ to $\alpha$, i.e., the behaviour of the absolute value of $\left(x_{n}-\alpha\right)$ for large values of $n$.

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• 1.II.12D

  Analysis I

  Part IA, 2001

  Explain what it means for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable on $[a, b]$, and give an example of a bounded function that is not Riemann integrable.

  Show each of the following statements is true for continuous functions $f$, but false for general Riemann integrable functions $f$.

  (i) If $f:[a, b] \rightarrow \mathbb{R}$ is such that $f(t) \geq 0$ for all $t$ in $[a, b]$ and $\int_{a}^{b} f(t) d t=0$, then $f(t)=0$ for all $t$ in $[a, b]$.

  (ii) $\int_{a}^{t} f(x) d x$ is differentiable and $\frac{d}{d t} \int_{a}^{t} f(x) d x=f(t)$.

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• 1.I $. 3 C$

  Analysis I

  Part IA, 2002

  Suppose $a_{n} \in \mathbb{R}$ for $n \geqslant 1$ and $a \in \mathbb{R}$. What does it mean to say that $a_{n} \rightarrow a$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$ ?

  Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$, then $1 / a_{n} \rightarrow 0$ as $n \rightarrow \infty$. Is the converse true? Give a proof or a counter example.

  Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow a$ with $a \neq 0$, then $1 / a_{n} \rightarrow 1 / a$ as $n \rightarrow \infty$.

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• 1.I.4C

  Analysis I

  Part IA, 2002

  Show that any bounded sequence of real numbers has a convergent subsequence.

  Give an example of a sequence of real numbers with no convergent subsequence.

  Give an example of an unbounded sequence of real numbers with a convergent subsequence.

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• 1.II.9C

  Analysis I

  Part IA, 2002

  State some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.

  In each of the following cases state whether $\sum_{n=1}^{\infty} a_{n}$ converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.

  (i) $a_{n}=(-1)^{n}(\log (n+1))^{-1}$.

  (ii) $a_{2 n}=(2 n)^{-2}, a_{2 n-1}=-n^{-2}$.

  (iii) $a_{3 n-2}=-(2 n-1)^{-1}, a_{3 n-1}=(4 n-1)^{-1}, a_{3 n}=(4 n)^{-1}$.

  (iv) $a_{2^{n}+r}=(-1)^{n}\left(2^{n}+r\right)^{-1}$ for $0 \leqslant r \leqslant 2^{n}-1, n \geqslant 0$.

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• 1.II.10C

  Analysis I

  Part IA, 2002

  Show that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.

  Write down examples of the following functions (no proof is required).

  (i) A continuous function $f_{1}:(0,1) \rightarrow \mathbb{R}$ which is not bounded.

  (ii) A continuous function $f_{2}:(0,1) \rightarrow \mathbb{R}$ which is bounded but does not attain its bounds.

  (iii) A bounded function $f_{3}:[0,1] \rightarrow \mathbb{R}$ which is not continuous.

  (iv) A function $f_{4}:[0,1] \rightarrow \mathbb{R}$ which is not bounded on any interval $[a, b]$ with $0 \leqslant a<b \leqslant 1 .$

  [Hint: Consider first how to define $f_{4}$ on the rationals.]

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• 1.II.11C

  Analysis I

  Part IA, 2002

  State the mean value theorem and deduce it from Rolle's theorem.

  Use the mean value theorem to show that, if $h: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable with $h^{\prime}(x)=0$ for all $x$, then $h$ is constant.

  By considering the derivative of the function $g$ given by $g(x)=e^{-a x} f(x)$, find all the solutions of the differential equation $f^{\prime}(x)=a f(x)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and $a$ is a fixed real number.

  Show that, if $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then the function $F: \mathbb{R} \rightarrow \mathbb{R}$ given by

  $$F(x)=\int_{0}^{x} f(t) d t$$

  is differentiable with $F^{\prime}(x)=f(x)$.

  Find the solution of the equation

  $$g(x)=A+\int_{0}^{x} g(t) d t$$

  where $g: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and $A$ is a real number. You should explain why the solution is unique.

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• 1.II.12C

  Analysis I

  Part IA, 2002

  Prove Taylor's theorem with some form of remainder.

  An infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the differential equation

  $$f^{(3)}(x)=f(x)$$

  and the conditions $f(0)=1, f^{\prime}(0)=f^{\prime \prime}(0)=0$. If $R>0$ and $j$ is a positive integer, explain why we can find an $M_{j}$ such that

  $$\left|f^{(j)}(x)\right| \leqslant M_{j}$$

  for all $x$ with $|x| \leqslant R$. Explain why we can find an $M$ such that

  $$\left|f^{(j)}(x)\right| \leqslant M$$

  for all $x$ with $|x| \leqslant R$ and all $j \geqslant 0$.

  Use your form of Taylor's theorem to show that

  $$f(x)=\sum_{n=0}^{\infty} \frac{x^{3 n}}{(3 n) !}$$

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• 1.I $. 3 \mathrm{~F} \quad$

  Analysis I

  Part IA, 2008

  State the ratio test for the convergence of a series.

  Find all real numbers $x$ such that the series

  $$\sum_{n=1}^{\infty} \frac{x^{n}-1}{n}$$

  converges.

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• 1.I.4E

  Analysis I

  Part IA, 2008

  Let $f:[0,1] \rightarrow \mathbb{R}$ be Riemann integrable, and for $0 \leqslant x \leqslant 1$ set $F(x)=\int_{0}^{x} f(t) d t$.

  Assuming that $f$ is continuous, prove that for every $0<x<1$ the function $F$ is differentiable at $x$, with $F^{\prime}(x)=f(x)$.

  If we do not assume that $f$ is continuous, must it still be true that $F$ is differentiable at every $0<x<1$ ? Justify your answer.

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• 1.II $. 9 \mathrm{~F} \quad$

  Analysis I

  Part IA, 2008

  Investigate the convergence of the series (i) $\sum_{n=2}^{\infty} \frac{1}{n^{p}(\log n)^{q}}$ (ii) $\sum_{n=3}^{\infty} \frac{1}{n(\log \log n)^{r}}$

  for positive real values of $p, q$ and $r$.

  [You may assume that for any positive real value of $\alpha, \log n<n^{\alpha}$ for $n$ sufficiently large. You may assume standard tests for convergence, provided that they are clearly stated.]

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• 1.II.10D

  Analysis I

  Part IA, 2008

  (a) State and prove the intermediate value theorem.

  (b) An interval is a subset $I$ of $\mathbb{R}$ with the property that if $x$ and $y$ belong to $I$ and $x<z<y$ then $z$ also belongs to $I$. Prove that if $I$ is an interval and $f$ is a continuous function from $I$ to $\mathbb{R}$ then $f(I)$ is an interval.

  (c) For each of the following three pairs $(I, J)$ of intervals, either exhibit a continuous function $f$ from $I$ to $\mathbb{R}$ such that $f(I)=J$ or explain briefly why no such continuous function exists: (i) $I=[0,1], \quad J=[0, \infty)$; (ii) $I=(0,1], \quad J=[0, \infty)$; (iii) $I=(0,1], \quad J=(-\infty, \infty)$.

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• 1.II.11D

  Analysis I

  Part IA, 2008

  (a) Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$ and suppose that both $f$ and $g$ are differentiable at the real number $x$. Prove that the product $f g$ is also differentiable at $x$.

  (b) Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and let $g(x)=x^{2} f(x)$ for every $x$. Prove that $g$ is differentiable at $x$ if and only if either $x=0$ or $f$ is differentiable at $x$.

  (c) Now let $f$ be any continuous function from $\mathbb{R}$ to $\mathbb{R}$ and let $g(x)=f(x)^{2}$ for every $x$. Prove that $g$ is differentiable at $x$ if and only if at least one of the following two possibilities occurs:

  (i) $f$ is differentiable at $x$;

  (ii) $f(x)=0$ and

  $$\frac{f(x+h)}{|h|^{1 / 2}} \longrightarrow 0 \quad \text { as } \quad h \rightarrow 0$$

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• 1.II.12E

  Analysis I

  Part IA, 2008

  Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a complex power series. Prove that there exists an $R \in[0, \infty]$ such that the series converges for every $z$ with $|z|<R$ and diverges for every $z$ with $|z|>R$.

  Find the value of $R$ for each of the following power series: (i) $\sum_{n=1}^{\infty} \frac{1}{n^{2}} z^{n}$; (ii) $\sum_{n=0}^{\infty} z^{n !}$.

  In each case, determine at which points on the circle $|z|=R$ the series converges.

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• Paper 1, Section I, $3 \mathbf{F}$

  Analysis I

  Part IA, 2009

  Determine the limits as $n \rightarrow \infty$ of the following sequences:

  (a) $a_{n}=n-\sqrt{n^{2}-n}$;

  (b) $b_{n}=\cos ^{2}\left(\pi \sqrt{n^{2}+n}\right)$.

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2009

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of complex numbers. Prove that there exists $R \in[0, \infty]$ such that the power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

  Give an example of a power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ that diverges if $z=\pm 1$ and converges if $z=\pm \mathrm{i}$.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2009

  For each of the following series, determine for which real numbers $x$ it diverges, for which it converges, and for which it converges absolutely. Justify your answers briefly.

  (a) $\sum_{n \geqslant 1} \frac{3+(\sin x)^{n}}{n}(\sin x)^{n}$,

  (b) $\quad \sum_{n \geqslant 1}|\sin x|^{n} \frac{(-1)^{n}}{\sqrt{n}}$,

  (c)

  $$\sum_{n \geqslant 1} \underbrace{\sin (0.99 \sin (0.99 \ldots \sin (0.99 x) \ldots))}_{n \text { times }} .$$

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2009

  State and prove the intermediate value theorem.

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and let $P=(a, b)$ be a point of the plane $\mathbb{R}^{2}$. Show that the set of distances from points $(x, f(x))$ on the graph of $f$ to the point $P$ is an interval $[A, \infty)$ for some value $A \geqslant 0$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2009

  State and prove Rolle's theorem.

  Let $f$ and $g$ be two continuous, real-valued functions on a closed, bounded interval $[a, b]$ that are differentiable on the open interval $(a, b)$. By considering the determinant

  $$\phi(x)=\left|\begin{array}{ccc} 1 & 1 & 0 \\ f(a) & f(b) & f(x) \\ g(a) & g(b) & g(x) \end{array}\right|=g(x)(f(b)-f(a))-f(x)(g(b)-g(a))$$

  or otherwise, show that there is a point $c \in(a, b)$ with

  $$f^{\prime}(c)(g(b)-g(a))=g^{\prime}(c)(f(b)-f(a))$$

  Suppose that $f, g:(0, \infty) \rightarrow \mathbb{R}$ are differentiable functions with $f(x) \rightarrow 0$ and $g(x) \rightarrow 0$ as $x \rightarrow 0$. Prove carefully that if the $\operatorname{limit}_{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\ell$ exists and is finite, then the limit $\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}$ also exists and equals $\ell$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2009

  (a) What does it mean for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable?

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function. Suppose that for every $\delta>0$ there is a sequence

  $$0 \leqslant a_{1}<b_{1} \leqslant a_{2}<b_{2} \leqslant \ldots \leqslant a_{n}<b_{n} \leqslant 1$$

  such that for each $i$ the function $f$ is Riemann integrable on the closed interval $\left[a_{i}, b_{i}\right]$, and such that $\sum_{i=1}^{n}\left(b_{i}-a_{i}\right) \geqslant 1-\delta$. Prove that $f$ is Riemann integrable on $[0,1]$.

  (c) Let $f:[0,1] \rightarrow \mathbb{R}$ be defined as follows. We set $f(x)=1$ if $x$ has an infinite decimal expansion that consists of 2 s and $7 \mathrm{~s}$ only, and otherwise we set $f(x)=0$. Prove that $f$ is Riemann integrable and determine $\int_{0}^{1} f(x) \mathrm{d} x$.

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• Paper 1, Section I, D

  Analysis I

  Part IA, 2010

  Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $R \in[0, \infty]$.

  Suppose the power series has radius of convergence $R$, with $0<R<\infty$. Show that the sequence $\left|a_{n} z^{n}\right|$ is unbounded if $|z|>R$.

  Find the radius of convergence of $\sum_{n \geqslant 1} z^{n} / n^{3}$.

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2010

  Find the limit of each of the following sequences; justify your answers.

  (i)

  $$\frac{1+2+\ldots+n}{n^{2}}$$

  (ii)

  $$\sqrt[n]{n}$$

  (iii)

  $$\left(a^{n}+b^{n}\right)^{1 / n} \quad \text { with } \quad 0<a \leqslant b$$

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2010

  Determine whether the following series converge or diverge. Any tests that you use should be carefully stated.

  (a)

  $$\sum_{n \geqslant 1} \frac{n !}{n^{n}}$$

  (b)

  $$\sum_{n \geqslant 1} \frac{1}{n+(\log n)^{2}}$$

  (c)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{1+\sqrt{n}}$$

  (d)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{n\left(2+(-1)^{n}\right)}$$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2010

  (a) State and prove Taylor's theorem with the remainder in Lagrange's form.

  (b) Suppose that $e: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $e(0)=1$ and $e^{\prime}(x)=e(x)$ for all $x \in \mathbb{R}$. Use the result of (a) to prove that

  $$e(x)=\sum_{n \geqslant 0} \frac{x^{n}}{n !} \quad \text { for all } \quad x \in \mathbb{R}$$

  [No property of the exponential function may be assumed.]

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2010

  Define what it means for a bounded function $f:[a, \infty) \rightarrow \mathbb{R}$ to be Riemann integrable.

  Show that a monotonic function $f:[a, b] \rightarrow \mathbb{R}$ is Riemann integrable, where $-\infty<a<b<\infty$.

  Prove that if $f:[1, \infty) \rightarrow \mathbb{R}$ is a decreasing function with $f(x) \rightarrow 0$ as $x \rightarrow \infty$, then $\sum_{n \geqslant 1} f(n)$ and $\int_{1}^{\infty} f(x) d x$ either both diverge or both converge.

  Hence determine, for $\alpha \in \mathbb{R}$, when $\sum_{n \geqslant 1} n^{\alpha}$ converges.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2010

  (a) Let $n \geqslant 1$ and $f$ be a function $\mathbb{R} \rightarrow \mathbb{R}$. Define carefully what it means for $f$ to be $n$ times differentiable at a point $x_{0} \in \mathbb{R}$.

  $$\text { Set } \operatorname{sign}(x)= \begin{cases}x /|x|, & x \neq 0 \\ 0, & x=0 .\end{cases}$$

  Consider the function $f(x)$ on the real line, with $f(0)=0$ and

  $$f(x)=x^{2} \operatorname{sign}(x)\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 .$$

  (b) Is $f(x)$ differentiable at $x=0$ ?

  (c) Show that $f(x)$ has points of non-differentiability in any neighbourhood of $x=0$.

  (d) Prove that, in any finite interval $I$, the derivative $f^{\prime}(x)$, at the points $x \in I$ where it exists, is bounded: $\left|f^{\prime}(x)\right| \leqslant C$ where $C$ depends on $I$.

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• Paper 1, Section I, $3 F$

  Analysis I

  Part IA, 2011

  (a) State, without proof, the Bolzano-Weierstrass Theorem.

  (b) Give an example of a sequence that does not have a convergent subsequence.

  (c) Give an example of an unbounded sequence having a convergent subsequence.

  (d) Let $a_{n}=1+(-1)^{\lfloor n / 2\rfloor}(1+1 / n)$, where $\lfloor x\rfloor$ denotes the integer part of $x$. Find all values $c$ such that the sequence $\left\{a_{n}\right\}$ has a subsequence converging to $c$. For each such value, provide a subsequence converging to it.

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• Paper 1, Section I, D

  Analysis I

  Part IA, 2011

  Find the radius of convergence of each of the following power series. (i) $\sum_{n \geqslant 1} n^{2} z^{n}$ (ii) $\sum_{n \geqslant 1} n^{n^{1 / 3}} z^{n}$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2011

  (a) State, without proof, the ratio test for the series $\sum_{n \geqslant 1} a_{n}$, where $a_{n}>0$. Give examples, without proof, to show that, when $a_{n+1}<a_{n}$ and $a_{n+1} / a_{n} \rightarrow 1$, the series may converge or diverge.

  (b) Prove that $\sum_{k=1}^{n-1} \frac{1}{k} \geqslant \log n$.

  (c) Now suppose that $a_{n}>0$ and that, for $n$ large enough, $\frac{a_{n+1}}{a_{n}} \leqslant 1-\frac{c}{n}$ where $c>1$. Prove that the series $\sum_{n \geqslant 1} a_{n}$ converges.

  [You may find it helpful to prove the inequality $\log (1-x)<-x$ for $0<x<1$.]

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2011

  State and prove the Intermediate Value Theorem.

  A fixed point of a function $f: X \rightarrow X$ is an $x \in X$ with $f(x)=x$. Prove that every continuous function $f:[0,1] \rightarrow[0,1]$ has a fixed point.

  Answer the following questions with justification.

  (i) Does every continuous function $f:(0,1) \rightarrow(0,1)$ have a fixed point?

  (ii) Does every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ have a fixed point?

  (iii) Does every function $f:[0,1] \rightarrow[0,1]$ (not necessarily continuous) have a fixed point?

  (iv) Let $f:[0,1] \rightarrow[0,1]$ be a continuous function with $f(0)=1$ and $f(1)=0$. Can $f$ have exactly two fixed points?

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2011

  For each of the following two functions $f: \mathbb{R} \rightarrow \mathbb{R}$, determine the set of points at which $f$ is continuous, and also the set of points at which $f$ is differentiable.

  $$\begin{aligned} &\text { (i) } f(x)= \begin{cases}x & \text { if } x \in \mathbb{Q} \\ -x & \text { if } x \notin \mathbb{Q}\end{cases} \\ &\text { (ii) } f(x)= \begin{cases}x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} \end{aligned}$$

  By modifying the function in (i), or otherwise, find a function (not necessarily continuous) $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is differentiable at 0 and nowhere else.

  Find a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is not differentiable at the points $1 / 2,1 / 3,1 / 4, \ldots$, but is differentiable at all other points.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2011

  State and prove the Fundamental Theorem of Calculus.

  Let $f:[0,1] \rightarrow \mathbb{R}$ be integrable, and set $F(x)=\int_{0}^{x} f(t) \mathrm{d} t$ for $0<x<1$. Must $F$ be differentiable?

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must the Riemann integral of $g$ from 0 to 1 exist?

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2012

  What does it mean to say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x_{0} \in \mathbb{R}$ ?

  Give an example of a continuous function $f:(0,1] \rightarrow \mathbb{R}$ which is bounded but attains neither its upper bound nor its lower bound.

  The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and non-negative, and satisfies $f(x) \rightarrow 0$ as $x \rightarrow \infty$ and $f(x) \rightarrow 0$ as $x \rightarrow-\infty$. Show that $f$ is bounded above and attains its upper bound.

  [Standard results about continuous functions on closed bounded intervals may be used without proof if clearly stated.]

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• Paper 1, Section I, $4 \mathbf{F}$

  Analysis I

  Part IA, 2012

  Let $f, g:[0,1] \rightarrow \mathbb{R}$ be continuous functions with $g(x) \geqslant 0$ for $x \in[0,1]$. Show that

  $$\int_{0}^{1} f(x) g(x) d x \leqslant M \int_{0}^{1} g(x) d x$$

  where $M=\sup \{|f(x)|: x \in[0,1]\}$.

  Prove there exists $\alpha \in[0,1]$ such that

  $$\int_{0}^{1} f(x) g(x) d x=f(\alpha) \int_{0}^{1} g(x) d x$$

  [Standard results about continuous functions and their integrals may be used without proof, if clearly stated.]

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2012

  (a) What does it mean to say that the sequence $\left(x_{n}\right)$ of real numbers converges to $\ell \in \mathbb{R} ?$

  Suppose that $\left(y_{n}^{(1)}\right),\left(y_{n}^{(2)}\right), \ldots,\left(y_{n}^{(k)}\right)$ are sequences of real numbers converging to the same limit $\ell$. Let $\left(x_{n}\right)$ be a sequence such that for every $n$,

  $$x_{n} \in\left\{y_{n}^{(1)}, y_{n}^{(2)}, \ldots, y_{n}^{(k)}\right\}$$

  Show that $\left(x_{n}\right)$ also converges to $\ell$.

  Find a collection of sequences $\left(y_{n}^{(j)}\right), j=1,2, \ldots$ such that for every $j,\left(y_{n}^{(j)}\right) \rightarrow \ell$ but the sequence $\left(x_{n}\right)$ defined by $x_{n}=y_{n}^{(n)}$ diverges.

  (b) Let $a, b$ be real numbers with $0<a<b$. Sequences $\left(a_{n}\right),\left(b_{n}\right)$ are defined by $a_{1}=a, b_{1}=b$ and

  $$\text { for all } n \geqslant 1, \quad a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2} \text {. }$$

  Show that $\left(a_{n}\right)$ and $\left(b_{n}\right)$ converge to the same limit.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2012

  Let $\left(a_{n}\right)$ be a sequence of reals.

  (i) Show that if the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent then so is the sequence $\left(\frac{a_{n}}{n}\right)$.

  (ii) Give an example to show the sequence $\left(\frac{a_{n}}{n}\right)$ being convergent does not imply that the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent.

  (iii) If $a_{n+k}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for each positive integer $k$, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

  (iv) If $a_{n+f(n)}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for every function $f$ from the positive integers to the positive integers, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2012

  Let $f$ be a continuous function from $(0,1)$ to $(0,1)$ such that $f(x)<x$ for every $0<x<1$. We write $f^{n}$ for the $n$-fold composition of $f$ with itself (so for example $\left.f^{2}(x)=f(f(x))\right)$.

  (i) Prove that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

  (ii) Must it be the case that for every $\epsilon>0$ there exists $n$ with the property that $f^{n}(x)<\epsilon$ for all $0<x<1$ ? Justify your answer.

  Now suppose that we remove the condition that $f$ be continuous.

  (iii) Give an example to show that it need not be the case that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

  (iv) Must it be the case that for some $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answer.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2012

  (a) (i) State the ratio test for the convergence of a real series with positive terms.

  (ii) Define the radius of convergence of a real power series $\sum_{n=0}^{\infty} a_{n} x^{n}$.

  (iii) Prove that the real power series $f(x)=\sum_{n} a_{n} x^{n}$ and $g(x)=\sum_{n}(n+1) a_{n+1} x^{n}$ have equal radii of convergence.

  (iv) State the relationship between $f(x)$ and $g(x)$ within their interval of convergence.

  (b) (i) Prove that the real series

  $$f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}, \quad g(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}$$

  have radius of convergence $\infty$.

  (ii) Show that they are differentiable on the real line $\mathbb{R}$, with $f^{\prime}=-g$ and $g^{\prime}=f$, and deduce that $f(x)^{2}+g(x)^{2}=1$.

  [You may use, without proof, general theorems about differentiating within the interval of convergence, provided that you give a clear statement of any such theorem.]

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• Paper 1, Section I, D

  Analysis I

  Part IA, 2013

  Show that $\exp (x) \geqslant 1+x$ for $x \geqslant 0$.

  Let $\left(a_{j}\right)$ be a sequence of positive real numbers. Show that for every $n$,

  $$\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)$$

  Deduce that $\prod_{1}^{n}\left(1+a_{j}\right)$ tends to a limit as $n \rightarrow \infty$ if and only if $\sum_{1}^{n} a_{j}$ does.

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2013

  (a) Suppose $b_{n} \geqslant b_{n+1} \geqslant 0$ for $n \geqslant 1$ and $b_{n} \rightarrow 0$. Show that $\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}$ converges.

  (b) Does the series $\sum_{n=2}^{\infty} \frac{1}{n \log n}$ converge or diverge? Explain your answer.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2013

  (a) Determine the radius of convergence of each of the following power series:

  $$\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}$$

  (b) State Taylor's theorem.

  Show that

  $$(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}$$

  for all $x \in(0,1)$, where

  $$c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}$$

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2013

  (a) Let $f:[a, b] \rightarrow \mathbb{R}$. Suppose that for every sequence $\left(x_{n}\right)$ in $[a, b]$ with limit $y \in[a, b]$, the sequence $\left(f\left(x_{n}\right)\right)$ converges to $f(y)$. Show that $f$ is continuous at $y$.

  (b) State the Intermediate Value Theorem.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a function with $f(a)=c<f(b)=d$. We say $f$ is injective if for all $x, y \in[a, b]$ with $x \neq y$, we have $f(x) \neq f(y)$. We say $f$ is strictly increasing if for all $x, y$ with $x<y$, we have $f(x)<f(y)$.

  (i) Suppose $f$ is strictly increasing. Show that it is injective, and that if $f(x)<f(y)$ then $x<y .$

  (ii) Suppose $f$ is continuous and injective. Show that if $a<x<b$ then $c<f(x)<d$. Deduce that $f$ is strictly increasing.

  (iii) Suppose $f$ is strictly increasing, and that for every $y \in[c, d]$ there exists $x \in[a, b]$ with $f(x)=y$. Show that $f$ is continuous at $b$. Deduce that $f$ is continuous on $[a, b]$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2013

  (i) State (without proof) Rolle's Theorem.

  (ii) State and prove the Mean Value Theorem.

  (iii) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$ with $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$. Show that there exists $\xi \in(a, b)$ such that

  $$\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$

  Deduce that if moreover $f(a)=g(a)=0$, and the limit

  $$\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$$

  exists, then

  $$\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a$$

  (iv) Deduce that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable then for any $a \in \mathbb{R}$

  $$f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .$$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2013

  Fix a closed interval $[a, b]$. For a bounded function $f$ on $[a, b]$ and a dissection $\mathcal{D}$ of $[a, b]$, how are the lower sum $s(f, \mathcal{D})$ and upper sum $S(f, \mathcal{D})$ defined? Show that $s(f, \mathcal{D}) \leqslant S(f, \mathcal{D})$.

  Suppose $\mathcal{D}^{\prime}$ is a dissection of $[a, b]$ such that $\mathcal{D} \subseteq \mathcal{D}^{\prime}$. Show that

  $$s(f, \mathcal{D}) \leqslant s\left(f, \mathcal{D}^{\prime}\right) \text { and } S\left(f, \mathcal{D}^{\prime}\right) \leqslant S(f, \mathcal{D})$$

  By using the above inequalities or otherwise, show that if $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are two dissections of $[a, b]$ then

  $$s\left(f, \mathcal{D}_{1}\right) \leqslant S\left(f, \mathcal{D}_{2}\right)$$

  For a function $f$ and dissection $\mathcal{D}=\left\{x_{0}, \ldots, x_{n}\right\}$ let

  $$p(f, \mathcal{D})=\prod_{k=1}^{n}\left[1+\left(x_{k}-x_{k-1}\right) \inf _{x \in\left[x_{k-1}, x_{k}\right]} f(x)\right]$$

  If $f$ is non-negative and Riemann integrable, show that

  $$p(f, \mathcal{D}) \leqslant e^{\int_{a}^{b} f(x) d x} .$$

  [You may use without proof the inequality $e^{t} \geqslant t+1$ for all $t$.]

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• Paper 1, Section I, D

  Analysis I

  Part IA, 2014

  Show that every sequence of real numbers contains a monotone subsequence.

  comment

• Paper 1, Section I, $4 \mathbf{F}$

  Analysis I

  Part IA, 2014

  Find the radius of convergence of the following power series: (i) $\sum_{n \geqslant 1} \frac{n !}{n^{n}} z^{n}$; (ii) $\sum_{n \geqslant 1} n^{n} z^{n !}$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2014

  (a) Show that for all $x \in \mathbb{R}$,

  $$\lim _{k \rightarrow \infty} 3^{k} \sin \left(x / 3^{k}\right)=x,$$

  stating carefully what properties of sin you are using.

  Show that the series $\sum_{n \geqslant 1} 2^{n} \sin \left(x / 3^{n}\right)$ converges absolutely for all $x \in \mathbb{R}$.

  (b) Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}, \theta$ not a multiple of $2 \pi$, the series

  $$\sum_{n \geqslant 1} a_{n} e^{i n \theta}$$

  converges.

  Hence, or otherwise, show that $\sum_{n \geqslant 1} \frac{\sin (n \theta)}{n}$ converges for all $\theta \in \mathbb{R}$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2014

  (i) State the Mean Value Theorem. Use it to show that if $f:(a, b) \rightarrow \mathbb{R}$ is a differentiable function whose derivative is identically zero, then $f$ is constant.

  (ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function and $\alpha>0$ a real number such that for all $x, y \in \mathbb{R}$,

  $$|f(x)-f(y)| \leqslant|x-y|^{\alpha} .$$

  Show that $f$ is continuous. Show moreover that if $\alpha>1$ then $f$ is constant.

  (iii) Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$. Assume also that the right derivative of $f$ at $a$ exists; that is, the limit

  $$\lim _{x \rightarrow a+} \frac{f(x)-f(a)}{x-a}$$

  exists. Show that for any $\epsilon>0$ there exists $x \in(a, b)$ satisfying

  $$\left|\frac{f(x)-f(a)}{x-a}-f^{\prime}(x)\right|<\epsilon .$$

  [You should not assume that $f^{\prime}$ is continuous.]

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2014

  (i) Prove Taylor's Theorem for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ differentiable $n$ times, in the following form: for every $x \in \mathbb{R}$ there exists $\theta$ with $0<\theta<1$ such that

  $$f(x)=\sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n)}(\theta x)}{n !} x^{n}$$

  [You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]

  (ii) The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and satisfies the differential equation $f^{\prime \prime}-f=0$ with $f(0)=A, f^{\prime}(0)=B$. Show that $f$ is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to $f$ at every point. Hence or otherwise show that for any $a, h \in \mathbb{R}$, the series

  $$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !} h^{k}$$

  converges to $f(a+h)$.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2014

  Define what it means for a function $f:[0,1] \rightarrow \mathbb{R}$ to be (Riemann) integrable. Prove that $f$ is integrable whenever it is

  (a) continuous,

  (b) monotonic.

  Let $\left\{q_{k}: k \in \mathbb{N}\right\}$ be an enumeration of all rational numbers in $[0,1)$. Define a function $f:[0,1] \rightarrow \mathbb{R}$ by $f(0)=0$,

  $$f(x)=\sum_{k \in Q(x)} 2^{-k}, \quad x \in(0,1]$$

  where

  $$Q(x)=\left\{k \in \mathbb{N}: q_{k} \in[0, x)\right\}$$

  Show that $f$ has a point of discontinuity in every interval $I \subset[0,1]$.

  Is $f$ integrable? [Justify your answer.]

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• Paper 1, Section I, $3 \mathbf{F}$

  Analysis I

  Part IA, 2015

  Find the following limits: (a) $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ (b) $\lim _{x \rightarrow 0}(1+x)^{1 / x}$ (c) $\lim _{x \rightarrow \infty} \frac{(1+x)^{\frac{x}{1+x}} \cos ^{4} x}{e^{x}}$

  Carefully justify your answers.

  [You may use standard results provided that they are clearly stated.]

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2015

  Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

  Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2015

  Let $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

  $$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$$

  Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

  Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

  Justify your answers.

  [You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

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• Paper 1, Section II, 10D

  Analysis I

  Part IA, 2015

  (a) For real numbers $a, b$ such that $a<b$, let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is bounded on $[a, b]$, and that $f$ attains its supremum and infimum on $[a, b]$.

  (b) For $x \in \mathbb{R}$, define

  $$g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.$$

  Find the set of points $x \in \mathbb{R}$ at which $g(x)$ is continuous.

  Does $g$ attain its supremum on $[0, \pi] ?$

  Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

  Justify your answers.

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• Paper 1, Section II, $11 D$

  Analysis I

  Part IA, 2015

  (i) State and prove the intermediate value theorem.

  (ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

  $$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$$

  or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2015

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

  Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

  [You may use the mean value theorem provided that it is clearly stated.]

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• Paper 1, Section I, D

  Analysis I

  Part IA, 2016

  What does it mean to say that a sequence of real numbers $\left(x_{n}\right)$ converges to $x$ ? Suppose that $\left(x_{n}\right)$ converges to $x$. Show that the sequence $\left(y_{n}\right)$ given by

  $$y_{n}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$

  also converges to $x$.

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2016

  Let $a_{n}$ be the number of pairs of integers $(x, y) \in \mathbb{Z}^{2}$ such that $x^{2}+y^{2} \leqslant n^{2}$. What is the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ ? [You may use the comparison test, provided you state it clearly.]

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2016

  State the Bolzano-Weierstrass theorem. Use it to show that a continuous function $f:[a, b] \rightarrow \mathbb{R}$ attains a global maximum; that is, there is a real number $c \in[a, b]$ such that $f(c) \geqslant f(x)$ for all $x \in[a, b]$.

  A function $f$ is said to attain a local maximum at $c \in \mathbb{R}$ if there is some $\varepsilon>0$ such that $f(c) \geqslant f(x)$ whenever $|x-c|<\varepsilon$. Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and that $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R}$. Show that there is at most one $c \in \mathbb{R}$ at which $f$ attains a local maximum.

  If there is a constant $K<0$ such that $f^{\prime \prime}(x)<K$ for all $x \in \mathbb{R}$, show that $f$ attains a global maximum. [Hint: if $g^{\prime}(x)<0$ for all $x \in \mathbb{R}$, then $g$ is decreasing.]

  Must $f: \mathbb{R} \rightarrow \mathbb{R}$ attain a global maximum if we merely require $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R} ?$ Justify your answer.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2016

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$. We say that $x \in \mathbb{R}$ is a real root of $f$ if $f(x)=0$. Show that if $f$ is differentiable and has $k$ distinct real roots, then $f^{\prime}$ has at least $k-1$ real roots. [Rolle's theorem may be used, provided you state it clearly.]

  Let $p(x)=\sum_{i=1}^{n} a_{i} x^{d_{i}}$ be a polynomial in $x$, where all $a_{i} \neq 0$ and $d_{i+1}>d_{i}$. (In other words, the $a_{i}$ are the nonzero coefficients of the polynomial, arranged in order of increasing power of $x$.) The number of sign changes in the coefficients of $p$ is the number of $i$ for which $a_{i} a_{i+1}<0$. For example, the polynomial $x^{5}-x^{3}-x^{2}+1$ has 2 sign changes. Show by induction on $n$ that the number of positive real roots of $p$ is less than or equal to the number of sign changes in its coefficients.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2016

  If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are sequences converging to $x$ and $y$ respectively, show that the sequence $\left(x_{n}+y_{n}\right)$ converges to $x+y$.

  If $x_{n} \neq 0$ for all $n$ and $x \neq 0$, show that the sequence $\left(\frac{1}{x_{n}}\right)$ converges to $\frac{1}{x}$.

  (a) Find $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$.

  (b) Determine whether $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}}$ converges.

  Justify your answers.

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• Paper 1, Section II, 12F

  Analysis I

  Part IA, 2016

  Let $f:[0,1] \rightarrow \mathbb{R}$ satisfy $|f(x)-f(y)| \leqslant|x-y|$ for all $x, y \in[0,1]$.

  Show that $f$ is continuous and that for all $\varepsilon>0$, there exists a piecewise constant function $g$ such that

  $$\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .$$

  For all integers $n \geqslant 1$, let $u_{n}=\int_{0}^{1} f(t) \cos (n t) d t$. Show that the sequence $\left(u_{n}\right)$ converges to 0 .

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2017

  Given an increasing sequence of non-negative real numbers $\left(a_{n}\right)_{n=1}^{\infty}$, let

  $$s_{n}=\frac{1}{n} \sum_{k=1}^{n} a_{k}$$

  Prove that if $s_{n} \rightarrow x$ as $n \rightarrow \infty$ for some $x \in \mathbb{R}$ then also $a_{n} \rightarrow x$ as $n \rightarrow \infty$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2017

  (a) Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a non-negative and decreasing sequence of real numbers. Prove that $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{k=0}^{\infty} 2^{k} x_{2^{k}}$ converges.

  (b) For $s \in \mathbb{R}$, prove that $\sum_{n=1}^{\infty} n^{-s}$ converges if and only if $s>1$.

  (c) For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} 2^{-n} n^{k}=0$$

  (d) The sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined by $a_{0}=1$ and $a_{n+1}=2^{a_{n}}$ for $n \geqslant 0$. For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} \frac{2^{n^{k}}}{a_{n}}=0$$

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• Paper 1, Section I, $4 \mathrm{E}$

  Analysis I

  Part IA, 2017

  Show that if the power series $\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C})$ converges for some fixed $z=z_{0}$, then it converges absolutely for every $z$ satisfying $|z|<\left|z_{0}\right|$.

  Define the radius of convergence of a power series.

  Give an example of $v \in \mathbb{C}$ and an example of $w \in \mathbb{C}$ such that $|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n}$ converges and $\sum_{n=1}^{\infty} \frac{w^{n}}{n}$ diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{z^{n}}{n}$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2017

  (a) State the Intermediate Value Theorem.

  (b) Define what it means for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to be differentiable at a point $a \in \mathbb{R}$. If $f$ is differentiable everywhere on $\mathbb{R}$, must $f^{\prime}$ be continuous everywhere? Justify your answer.

  State the Mean Value Theorem.

  (c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere. Let $a, b \in \mathbb{R}$ with $a<b$.

  If $f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b)$, prove that there exists $c \in[a, b]$ such that $f^{\prime}(c)=y$. [Hint: consider the function $g$ defined by

  $$g(x)=\frac{f(x)-f(a)}{x-a}$$

  if $x \neq a$ and $\left.g(a)=f^{\prime}(a) .\right]$

  If additionally $f(a) \leqslant 0 \leqslant f(b)$, deduce that there exists $d \in[a, b]$ such that $f^{\prime}(d)+f(d)=y$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2017

  Let $a, b \in \mathbb{R}$ with $a<b$ and let $f:(a, b) \rightarrow \mathbb{R}$.

  (a) Define what it means for $f$ to be continuous at $y_{0} \in(a, b)$.

  $f$ is said to have a local minimum at $c \in(a, b)$ if there is some $\varepsilon>0$ such that $f(c) \leqslant f(x)$ whenever $x \in(a, b)$ and $|x-c|<\varepsilon$.

  If $f$ has a local minimum at $c \in(a, b)$ and $f$ is differentiable at $c$, show that $f^{\prime}(c)=0$.

  (b) $f$ is said to be convex if

  $$f(\lambda x+(1-\lambda) y) \leqslant \lambda f(x)+(1-\lambda) f(y)$$

  for every $x, y \in(a, b)$ and $\lambda \in[0,1]$. If $f$ is convex, $r \in \mathbb{R}$ and $\left[y_{0}-|r|, y_{0}+|r|\right] \subset(a, b)$, prove that

  $$(1+\lambda) f\left(y_{0}\right)-\lambda f\left(y_{0}-r\right) \leqslant f\left(y_{0}+\lambda r\right) \leqslant(1-\lambda) f\left(y_{0}\right)+\lambda f\left(y_{0}+r\right)$$

  for every $\lambda \in[0,1]$.

  Deduce that if $f$ is convex then $f$ is continuous.

  If $f$ is convex and has a local minimum at $c \in(a, b)$, prove that $f$ has a global minimum at $c$, i.e., that $f(x) \geqslant f(c)$ for every $x \in(a, b)$. [Hint: argue by contradiction.] Must $f$ be differentiable at $c$ ? Justify your answer.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2017

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function defined on the closed, bounded interval $[a, b]$ of $\mathbb{R}$. Suppose that for every $\varepsilon>0$ there is a dissection $\mathcal{D}$ of $[a, b]$ such that $S_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon$, where $s_{\mathcal{D}}(f)$ and $S_{\mathcal{D}}(f)$ denote the lower and upper Riemann sums of $f$ for the dissection $\mathcal{D}$. Deduce that $f$ is Riemann integrable. [You may assume without proof that $s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f)$ for all dissections $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of $\left.[a, b] .\right]$

  Prove that if $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

  Let $g:(0,1] \rightarrow \mathbb{R}$ be a bounded continuous function. Show that for any $\lambda \in \mathbb{R}$, the function $f:[0,1] \rightarrow \mathbb{R}$ defined by

  $$f(x)= \begin{cases}g(x) & \text { if } 0<x \leqslant 1 \\ \lambda & \text { if } x=0\end{cases}$$

  is Riemann integrable.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative $f^{\prime}$ is (bounded and) Riemann integrable. Show that

  $$\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$$

  [You may use the Mean Value Theorem without proof.]

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2018

  Prove that an increasing sequence in $\mathbb{R}$ that is bounded above converges.

  Let $f: \mathbb{R} \rightarrow(0, \infty)$ be a decreasing function. Let $x_{1}=1$ and $x_{n+1}=x_{n}+f\left(x_{n}\right)$. Prove that $x_{n} \rightarrow \infty$ as $n \rightarrow \infty$.

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• Paper 1, Section I, $4 \mathrm{D}$

  Analysis I

  Part IA, 2018

  Define the radius of convergence $R$ of a complex power series $\sum a_{n} z^{n}$. Prove that $\sum a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

  If $\left|a_{n}\right| \leqslant\left|b_{n}\right|$ for all $n$ does it follow that the radius of convergence of $\sum a_{n} z^{n}$ is at least that of $\sum b_{n} z^{n}$ ? Justify your answer.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2018

  (a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function, and let $x \in \mathbb{R}$. Define what it means for $f$ to be continuous at $x$. Show that $f$ is continuous at $x$ if and only if $f\left(x_{n}\right) \rightarrow f(x)$ for every sequence $\left(x_{n}\right)$ with $x_{n} \rightarrow x$.

  (b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-constant polynomial. Show that its image $\{f(x): x \in \mathbb{R}\}$ is either the real line $\mathbb{R}$, the interval $[a, \infty)$ for some $a \in \mathbb{R}$, or the interval $(-\infty, a]$ for some $a \in \mathbb{R}$.

  (c) Let $\alpha>1$, let $f:(0, \infty) \rightarrow \mathbb{R}$ be continuous, and assume that $f(x)=f\left(x^{\alpha}\right)$ holds for all $x>0$. Show that $f$ must be constant.

  Is this also true when the condition that $f$ be continuous is dropped?

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• Paper 1, Section II, $10 F$

  Analysis I

  Part IA, 2018

  (a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable at $x_{0} \in \mathbb{R}$. Show that $f$ is continuous at $x_{0}$.

  (b) State the Mean Value Theorem. Prove the following inequalities:

  $$\left|\cos \left(e^{-x}\right)-\cos \left(e^{-y}\right)\right| \leqslant|x-y| \quad \text { for } x, y \geqslant 0$$

  and

  $$\log (1+x) \leqslant \frac{x}{\sqrt{1+x}} \text { for } x \geqslant 0 .$$

  (c) Determine at which points the following functions from $\mathbb{R}$ to $\mathbb{R}$ are differentiable, and find their derivatives at the points at which they are differentiable:

  $$f(x)=\left\{\begin{array}{ll} |x|^{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0, \end{array} \quad g(x)=\cos (|x|), \quad h(x)=x|x|\right.$$

  (d) Determine the points at which the following function from $\mathbb{R}$ to $\mathbb{R}$ is continuous:

  $$f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \text { or } x=0 \\ 1 / q & \text { if } x=p / q \text { where } p \in \mathbb{Z} \backslash\{0\} \text { and } q \in \mathbb{N} \text { are relatively prime. }\end{cases}$$

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2018

  State and prove the Comparison Test for real series.

  Assume $0 \leqslant x_{n}<1$ for all $n \in \mathbb{N}$. Show that if $\sum x_{n}$ converges, then so do $\sum x_{n}^{2}$ and $\sum \frac{x_{n}}{1-x_{n}}$. In each case, does the converse hold? Justify your answers.

  Let $\left(x_{n}\right)$ be a decreasing sequence of positive reals. Show that if $\sum x_{n}$ converges, then $n x_{n} \rightarrow 0$ as $n \rightarrow \infty$. Does the converse hold? If $\sum x_{n}$ converges, must it be the case that $(n \log n) x_{n} \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answers.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2018

  (a) Let $q_{1}, q_{2}, \ldots$ be a fixed enumeration of the rationals in $[0,1]$. For positive reals $a_{1}, a_{2}, \ldots$, define a function $f$ from $[0,1]$ to $\mathbb{R}$ by setting $f\left(q_{n}\right)=a_{n}$ for each $n$ and $f(x)=0$ for $x$ irrational. Prove that if $a_{n} \rightarrow 0$ then $f$ is Riemann integrable. If $a_{n} \rightarrow 0$, can $f$ be Riemann integrable? Justify your answer.

  (b) State and prove the Fundamental Theorem of Calculus.

  Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must $g$ be Riemann integrable on $[0,1]$ ?

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2019

  State the Bolzano-Weierstrass theorem.

  Let $\left(a_{n}\right)$ be a sequence of non-zero real numbers. Which of the following conditions is sufficient to ensure that $\left(1 / a_{n}\right)$ converges? Give a proof or counter-example as appropriate.

  (i) $a_{n} \rightarrow \ell$ for some real number $\ell$.

  (ii) $a_{n} \rightarrow \ell$ for some non-zero real number $\ell$.

  (iii) $\left(a_{n}\right)$ has no convergent subsequence.

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2019

  Let $\sum_{n=1}^{\infty} a_{n} x^{n}$ be a real power series that diverges for at least one value of $x$. Show that there exists a non-negative real number $R$ such that $\sum_{n=1}^{\infty} a_{n} x^{n}$ converges absolutely whenever $|x|<R$ and diverges whenever $|x|>R$.

  Find, with justification, such a number $R$ for each of the following real power series:

  (i) $\sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}}$;

  (ii) $\sum_{n=1}^{\infty} x^{n}\left(1+\frac{1}{n}\right)^{n}$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2019

  Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a function that is continuous at at least one point $z \in \mathbb{R}$. Suppose further that $g$ satisfies

  $$g(x+y)=g(x)+g(y)$$

  for all $x, y \in \mathbb{R}$. Show that $g$ is continuous on $\mathbb{R}$.

  Show that there exists a constant $c$ such that $g(x)=c x$ for all $x \in \mathbb{R}$.

  Suppose that $h: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function defined on $\mathbb{R}$ and that $h$ satisfies the equation

  $$h(x+y)=h(x) h(y)$$

  for all $x, y \in \mathbb{R}$. Show that $h$ is either identically zero or everywhere positive. What is the general form for $h$ ?

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2019

  State and prove the Intermediate Value Theorem.

  State the Mean Value Theorem.

  Suppose that the function $g$ is differentiable everywhere in some open interval containing $[a, b]$, and that $g^{\prime}(a)<k<g^{\prime}(b)$. By considering the functions $h$ and $f$ defined by

  $$h(x)=\frac{g(x)-g(a)}{x-a}(a<x \leqslant b), \quad h(a)=g^{\prime}(a)$$

  and

  $$f(x)=\frac{g(b)-g(x)}{b-x}(a \leqslant x<b), \quad f(b)=g^{\prime}(b),$$

  or otherwise, show that there is a subinterval $[\alpha, \beta] \subseteq[a, b]$ such that

  $$\frac{g(\beta)-g(\alpha)}{\beta-\alpha}=k$$

  Deduce that there exists $c \in(a, b)$ with $g^{\prime}(c)=k$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2019

  Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of positive real numbers. Let $s_{n}=\sum_{i=1}^{n} a_{i}$.

  (a) Show that if $\sum a_{n}$ and $\sum b_{n}$ converge then so does $\sum\left(a_{n}^{2}+b_{n}^{2}\right)^{1 / 2}$.

  (b) Show that if $\sum a_{n}$ converges then $\sum \sqrt{a_{n} a_{n+1}}$ converges. Is the converse true?

  (c) Show that if $\sum a_{n}$ diverges then $\sum \frac{a_{n}}{s_{n}}$ diverges. Is the converse true?

  $[$ For part (c), it may help to show that for any $N \in \mathbb{N}$ there exist $m \geqslant n \geqslant N$ with

  $$\left.\frac{a_{n+1}}{s_{n+1}}+\frac{a_{n+2}}{s_{n+2}}+\ldots+\frac{a_{m}}{s_{m}} \geqslant \frac{1}{2} .\right]$$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2019

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function. Define the upper and lower integrals of $f$. What does it mean to say that $f$ is Riemann integrable? If $f$ is Riemann integrable, what is the Riemann integral $\int_{0}^{1} f(x) d x$ ?

  Which of the following functions $f:[0,1] \rightarrow \mathbb{R}$ are Riemann integrable? For those that are Riemann integrable, find $\int_{0}^{1} f(x) d x$. Justify your answers.

  (i) $f(x)= \begin{cases}1 & \text { if } x \in \mathbb{Q} \\ 0 & \text { if } x \notin \mathbb{Q}\end{cases}$

  (ii) $f(x)=\left\{\begin{array}{ll}1 & \text { if } x \in A \\ 0 & \text { if } x \notin A\end{array}\right.$,

  where $A=\{x \in[0,1]: x$ has a base-3 expansion containing a 1$\}$;

  [Hint: You may find it helpful to note, for example, that $\frac{2}{3} \in A$ as one of the base-3 expansions of $\frac{2}{3}$ is $\left.0.1222 \ldots .\right]$

  (iii) $f(x)=\left\{\begin{array}{ll}1 & \text { if } x \in B \\ 0 & \text { if } x \notin B\end{array}\right.$,

  where $B=\{x \in[0,1]: x$ has a base $-3$ expansion containing infinitely many $1 \mathrm{~s}\}$.

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2020

  (a) Let $f$ be continuous in $[a, b]$, and let $g$ be strictly monotonic in $[\alpha, \beta]$, with a continuous derivative there, and suppose that $a=g(\alpha)$ and $b=g(\beta)$. Prove that

  $$\int_{a}^{b} f(x) d x=\int_{\alpha}^{\beta} f(g(u)) g^{\prime}(u) d u$$

  [Any version of the fundamental theorem of calculus may be used providing it is quoted correctly.]

  (b) Justifying carefully the steps in your argument, show that the improper Riemann integral

  $$\int_{0}^{e^{-1}} \frac{d x}{x\left(\log \frac{1}{x}\right)^{\theta}}$$

  converges for $\theta>1$, and evaluate it.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2020

  (a) State Rolle's theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is $N+1$ times differentiable and $x \in \mathbb{R}$ then

  $$f(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\ldots+\frac{f^{(N)}(0)}{N !} x^{N}+\frac{f^{(N+1)}(\theta x)}{(N+1) !} x^{N+1}$$

  for some $0<\theta<1$. Hence, or otherwise, show that if $f^{\prime}(x)=0$ for all $x \in \mathbb{R}$ then $f$ is constant.

  (b) Let $s: \mathbb{R} \rightarrow \mathbb{R}$ and $c: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that

  $$s^{\prime}(x)=c(x), \quad c^{\prime}(x)=-s(x), \quad s(0)=0 \quad \text { and } \quad c(0)=1$$

  Prove that (i) $s(x) c(a-x)+c(x) s(a-x)$ is independent of $x$, (ii) $s(x+y)=s(x) c(y)+c(x) s(y)$, (iii) $s(x)^{2}+c(x)^{2}=1$.

  Show that $c(1)>0$ and $c(2)<0$. Deduce there exists $1<k<2$ such that $s(2 k)=c(k)=0$ and $s(x+4 k)=s(x)$.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2020

  (a) Let $\left(x_{n}\right)$ be a bounded sequence of real numbers. Show that $\left(x_{n}\right)$ has a convergent subsequence.

  (b) Let $\left(z_{n}\right)$ be a bounded sequence of complex numbers. For each $n \geqslant 1$, write $z_{n}=x_{n}+i y_{n}$. Show that $\left(z_{n}\right)$ has a subsequence $\left(z_{n_{j}}\right)$ such that $\left(x_{n_{j}}\right)$ converges. Hence, or otherwise, show that $\left(z_{n}\right)$ has a convergent subsequence.

  (c) Write $\mathbb{N}=\{1,2,3, \ldots\}$ for the set of positive integers. Let $M$ be a positive real number, and for each $i \in \mathbb{N}$, let $X^{(i)}=\left(x_{1}^{(i)}, x_{2}^{(i)}, x_{3}^{(i)}, \ldots\right)$ be a sequence of real numbers with $\left|x_{j}^{(i)}\right| \leqslant M$ for all $i, j \in \mathbb{N}$. By induction on $i$ or otherwise, show that there exist sequences $N^{(i)}=\left(n_{1}^{(i)}, n_{2}^{(i)}, n_{3}^{(i)}, \ldots\right)$ of positive integers with the following properties:

  • for all $i \in \mathbb{N}$, the sequence $N^{(i)}$ is strictly increasing;

  • for all $i \in \mathbb{N}, N^{(i+1)}$ is a subsequence of $N^{(i)} ;$ and

  • for all $k \in \mathbb{N}$ and all $i \in \mathbb{N}$ with $1 \leqslant i \leqslant k$, the sequence

  $$\left(x_{n_{1}^{(k)}}^{(i)}, x_{n_{2}^{(k)}}^{(i)}, x_{n_{3}^{(k)}}^{(i)}, \ldots\right)$$

  converges.

  Hence, or otherwise, show that there exists a strictly increasing sequence $\left(m_{j}\right)$ of positive integers such that for all $i \in \mathbb{N}$ the sequence $\left(x_{m_{1}}^{(i)}, x_{m_{2}}^{(i)}, x_{m_{3}}^{(i)}, \ldots\right)$ converges.

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2021

  State and prove the alternating series test. Hence show that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges. Show also that

  $$\frac{7}{12} \leqslant \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \leqslant \frac{47}{60}$$

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• Paper 1, Section I, F

  Analysis I

  Part IA, 2021

  State and prove the Bolzano-Weierstrass theorem.

  Consider a bounded sequence $\left(x_{n}\right)$. Prove that if every convergent subsequence of $\left(x_{n}\right)$ converges to the same limit $L$ then $\left(x_{n}\right)$ converges to $L$.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2021

  (a) State the intermediate value theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous bijection and $x_{1}<x_{2}<x_{3}$ then either $f\left(x_{1}\right)<f\left(x_{2}\right)<f\left(x_{3}\right)$ or $f\left(x_{1}\right)>f\left(x_{2}\right)>f\left(x_{3}\right)$. Deduce that $f$ is either strictly increasing or strictly decreasing.

  (b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions. Which of the following statements are true, and which can be false? Give a proof or counterexample as appropriate.

  (i) If $f$ and $g$ are continuous then $f \circ g$ is continuous.

  (ii) If $g$ is strictly increasing and $f \circ g$ is continuous then $f$ is continuous.

  (iii) If $f$ is continuous and a bijection then $f^{-1}$ is continuous.

  (iv) If $f$ is differentiable and a bijection then $f^{-1}$ is differentiable.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2021

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function.

  (a) Let $m=\min _{x \in[a, b]} f(x)$ and $M=\max _{x \in[a, b]} f(x)$. If $g:[a, b] \rightarrow \mathbb{R}$ is a positive continuous function, prove that

  $$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$$

  directly from the definition of the Riemann integral.

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that

  $$\int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x \rightarrow f(0)$$

  as $n \rightarrow \infty$, and deduce that

  $$\int_{0}^{1} n f(x) e^{-n x} d x \rightarrow f(0)$$

  as $n \rightarrow \infty$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2021

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be $n$-times differentiable, for some $n>0$.

  (a) State and prove Taylor's theorem for $f$, with the Lagrange form of the remainder. [You may assume Rolle's theorem.]

  (b) Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an infinitely differentiable function such that $f(0)=1$ and $f^{\prime}(0)=0$, and satisfying the differential equation $f^{\prime \prime}(x)=-f(x)$. Prove carefully that

  $$f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !}$$

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2021

  (a) Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with $a_{n} \in \mathbb{C}$. Show that there exists $R \in[0, \infty]$ (called the radius of convergence) such that the series is absolutely convergent when $|z|<R$ but is divergent when $|z|>R$.

  Suppose that the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ is $R=2$. For a fixed positive integer $k$, find the radii of convergence of the following series. [You may assume that $\lim _{n \rightarrow \infty}\left|a_{n}\right|^{1 / n}$ exists.] (i) $\sum_{n=0}^{\infty} a_{n}^{k} z^{n}$. (ii) $\sum_{n=0}^{\infty} a_{n} z^{k n}$. (iii) $\sum_{n=0}^{\infty} a_{n} z^{n^{2}}$.

  (b) Suppose that there exist values of $z$ for which $\sum_{n=0}^{\infty} b_{n} e^{n z}$ converges and values for which it diverges. Show that there exists a real number $S$ such that $\sum_{n=0}^{\infty} b_{n} e^{n z}$ diverges whenever $\operatorname{Re}(z)>S$ and converges whenever $\operatorname{Re}(z)<S$.

  Determine the set of values of $z$ for which

  $$\sum_{n=0}^{\infty} \frac{2^{n} e^{i n z}}{(n+1)^{2}}$$

  converges.

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