Analysis

• 1.I.3B

  Analysis

  Part IA, 2003

  Define what it means for a function of a real variable to be differentiable at $x \in \mathbb{R}$.

  Prove that if a function is differentiable at $x \in \mathbb{R}$, then it is continuous there.

  Show directly from the definition that the function

  $$f(x)= \begin{cases}x^{2} \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{cases}$$

  is differentiable at 0 with derivative 0 .

  Show that the derivative $f^{\prime}(x)$ is not continuous at 0 .

  comment

• 1.I.4C

  Analysis

  Part IA, 2003

  Explain what is meant by the radius of convergence of a power series.

  Find the radius of convergence $R$ of each of the following power series: (i) $\sum_{n=1}^{\infty} n^{-2} z^{n}$, (ii) $\sum_{n=1}^{\infty}\left(n+\frac{1}{2^{n}}\right) z^{n}$.

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

  comment

• 1.II.9F

  Analysis

  Part IA, 2003

  Prove the Axiom of Archimedes.

  Let $x$ be a real number in $[0,1]$, and let $m, n$ be positive integers. Show that the limit

  $$\lim _{m \rightarrow \infty}\left[\lim _{n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right]$$

  exists, and that its value depends on whether $x$ is rational or irrational.

  [You may assume standard properties of the cosine function provided they are clearly stated.]

  comment

• 1.II.10F

  Analysis

  Part IA, 2003

  State without proof the Integral Comparison Test for the convergence of a series $\sum_{n=1}^{\infty} a_{n}$ of non-negative terms.

  Determine for which positive real numbers $\alpha$ the series $\sum_{n=1}^{\infty} n^{-\alpha}$ converges.

  In each of the following cases determine whether the series is convergent or divergent: (i) $\sum_{n=3}^{\infty} \frac{1}{n \log n}$, (ii) $\sum_{n=3}^{\infty} \frac{1}{(n \log n)(\log \log n)^{2}}$, (iii) $\sum_{n=3}^{\infty} \frac{1}{n^{(1+1 / n) \log n}}$.

  comment

• 1.II.11B

  Analysis

  Part IA, 2003

  Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Define the integral $\int_{a}^{b} f(x) d x$. (You are not asked to prove existence.)

  Suppose that $m, M$ are real numbers such that $m \leqslant f(x) \leqslant M$ for all $x \in[a, b]$. Stating clearly any properties of the integral that you require, show that

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

  The function $g:[a, b] \rightarrow \mathbb{R}$ is continuous and non-negative. Show that

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

  Now let $f$ be continuous on $[0,1]$. By suitable choice of $g$ show that

  $$\lim _{n \rightarrow \infty} \int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x=f(0),$$

  and by making an appropriate change of variable, or otherwise, show that

  $$\lim _{n \rightarrow \infty} \int_{0}^{1} n f(x) e^{-n x} d x=f(0) .$$

  comment

• 1.II.12C

  Analysis

  Part IA, 2003

  State carefully the formula for integration by parts for functions of a real variable.

  Let $f:(-1,1) \rightarrow \mathbb{R}$ be infinitely differentiable. Prove that for all $n \geqslant 1$ and all $t \in(-1,1)$,

  $f(t)=f(0)+f^{\prime}(0) t+\frac{1}{2 !} f^{\prime \prime}(0) t^{2}+\ldots+\frac{1}{(n-1) !} f^{(n-1)}(0) t^{n-1}+\frac{1}{(n-1) !} \int_{0}^{t} f^{(n)}(x)(t-x)^{n-1} d x .$

  By considering the function $f(x)=\log (1-x)$ at $x=1 / 2$, or otherwise, prove that the series

  $$\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}$$

  converges to $\log 2$.

  comment

• 1.I.3D

  Analysis

  Part IA, 2004

  Define the supremum or least upper bound of a non-empty set of real numbers.

  State the Least Upper Bound Axiom for the real numbers.

  Starting from the Least Upper Bound Axiom, show that if $\left(a_{n}\right)$ is a bounded monotonic sequence of real numbers, then it converges.

  comment

• 1.I.4E

  Analysis

  Part IA, 2004

  Let $f(x)=(1+x)^{1 / 2}$ for $x \in(-1,1)$. Show by induction or otherwise that for every integer $r \geq 1$,

  $$f^{(r)}(x)=(-1)^{r-1} \frac{(2 r-2) !}{2^{2 r-1}(r-1) !}(1+x)^{\frac{1}{2}-r}$$

  Evaluate the series

  $$\sum_{r=1}^{\infty}(-1)^{r-1} \frac{(2 r-2) !}{8^{r} r !(r-1) !}$$

  [You may use Taylor's Theorem in the form

  $$f(x)=f(0)+\sum_{r=1}^{n} \frac{f^{(r)}(0)}{r !} x^{r}+\int_{0}^{x} \frac{(x-t)^{n} f^{(n+1)}(t)}{n !} d t$$

  without proof.]

  comment

• 1.II.9D

  Analysis

  Part IA, 2004

  i) State Rolle's theorem.

  Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous functions which are differentiable on $(a, b)$.

  ii) Prove that for some $c \in(a, b)$,

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

  iii) Suppose that $f(a)=g(a)=0$, and that $\lim _{x \rightarrow a+} \frac{f^{\prime}(x)}{g^{\prime}(x)}$ exists and is equal to $L$.

  Prove that $\lim _{x \rightarrow a+} \frac{f(x)}{g(x)}$ exists and is also equal to $L$.

  [You may assume there exists a $\delta>0$ such that, for all $x \in(a, a+\delta), g^{\prime}(x) \neq 0$ and $g(x) \neq 0 .]$

  iv) Evaluate $\lim _{x \rightarrow 0} \frac{\log \cos x}{x^{2}}$.

  comment

• 1.II.10E

  Analysis

  Part IA, 2004

  Define, for an integer $n \geq 0$,

  $$I_{n}=\int_{0}^{\pi / 2} \sin ^{n} x d x$$

  Show that for every $n \geq 2, n I_{n}=(n-1) I_{n-2}$, and deduce that

  $$I_{2 n}=\frac{(2 n) !}{\left(2^{n} n !\right)^{2}} \frac{\pi}{2} \quad \text { and } \quad I_{2 n+1}=\frac{\left(2^{n} n !\right)^{2}}{(2 n+1) !}$$

  Show that $0<I_{n}<I_{n-1}$, and that

  $$\frac{2 n}{2 n+1}<\frac{I_{2 n+1}}{I_{2 n}}<1$$

  Hence prove that

  $$\lim _{n \rightarrow \infty} \frac{2^{4 n+1}(n !)^{4}}{(2 n+1)(2 n) !^{2}}=\pi .$$

  comment

• 1.II.11F

  Analysis

  Part IA, 2004

  Let $f$ be defined on $\mathbb{R}$, and assume that there exists at least one point $x_{0} \in \mathbb{R}$ at which $f$ is continuous. Suppose also that, for every $x, y \in \mathbb{R}, f$ satisfies the equation

  $$f(x+y)=f(x)+f(y)$$

  Show that $f$ is continuous on $\mathbb{R}$.

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

  Suppose that $g$ is a continuous function defined on $\mathbb{R}$ and that, for every $x, y \in \mathbb{R}$, $g$ satisfies the equation

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

  Show that if $g$ is not identically zero, then $g$ is everywhere positive. Find the general form of $g$.

  comment

• 1.II.12F

  Analysis

  Part IA, 2004

  (i) Show that if $a_{n}>0, b_{n}>0$ and

  $$\frac{a_{n+1}}{a_{n}} \leqslant \frac{b_{n+1}}{b_{n}}$$

  for all $n \geqslant 1$, and if $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

  (ii) Let

  $$c_{n}=\left(\begin{array}{c} 2 n \\ n \end{array}\right) 4^{-n} .$$

  By considering $\log c_{n}$, or otherwise, show that $c_{n} \rightarrow 0$ as $n \rightarrow \infty$.

  [Hint: $\log (1-x) \leqslant-x$ for $x \in(0,1)$.]

  (iii) Determine the convergence or otherwise of

  $$\sum_{n=1}^{\infty}\left(\begin{array}{c} 2 n \\ n \end{array}\right) x^{n}$$

  for (a) $x=\frac{1}{4}$, (b) $x=-\frac{1}{4}$.

  comment

• 1.I.3F

  Analysis

  Part IA, 2005

  Define the supremum or least upper bound of a non-empty set of real numbers.

  Let $A$ denote a non-empty set of real numbers which has a supremum but no maximum. Show that for every $\epsilon>0$ there are infinitely many elements of $A$ contained in the open interval

  $$(\sup A-\epsilon, \sup A) .$$

  Give an example of a non-empty set of real numbers which has a supremum and maximum and for which the above conclusion does not hold.

  comment

• 1.I.4D

  Analysis

  Part IA, 2005

  Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series in the complex plane with radius of convergence $R$. Show that $\left|a_{n} z^{n}\right|$ is unbounded in $n$ for any $z$ with $|z|>R$. State clearly any results on absolute convergence that are used.

  For every $R \in[0, \infty]$, show that there exists a power series $\sum_{n=0}^{\infty} a_{n} z^{n}$ with radius of convergence $R$.

  comment

• 1.II.9F

  Analysis

  Part IA, 2005

  Examine each of the following series and determine whether or not they converge.

  Give reasons in each case.

  $(i)$

  $(i i)$

  $$\sum_{n=1}^{\infty} \frac{1}{n^{2}+(-1)^{n+1} 2 n+1}$$

  (iii)

  $$\sum_{n=1}^{\infty} \frac{n^{3}+(-1)^{n} 8 n^{2}+1}{n^{4}+(-1)^{n+1} n^{2}}$$

  $(i v)$

  $$\sum_{n=1}^{\infty} \frac{n^{3}}{e^{e^{n}}}$$

  comment

• 1.II.10D

  Analysis

  Part IA, 2005

  Explain what it means for a bounded function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable.

  Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a strictly decreasing continuous function. Show that for each $x \in(0, \infty)$, there exists a unique point $g(x) \in(0, x)$ such that

  $$\frac{1}{x} \int_{0}^{x} f(t) d t=f(g(x)) .$$

  Find $g(x)$ if $f(x)=e^{-x}$.

  Suppose now that $f$ is differentiable and $f^{\prime}(x)<0$ for all $x \in(0, \infty)$. Prove that $g$ is differentiable at all $x \in(0, \infty)$ and $g^{\prime}(x)>0$ for all $x \in(0, \infty)$, stating clearly any results on the inverse of $f$ you use.

  comment

• 1.II.11E

  Analysis

  Part IA, 2005

  Prove that if $f$ is a continuous function on the interval $[a, b]$ with $f(a)<0<f(b)$ then $f(c)=0$ for some $c \in(a, b)$.

  Let $g$ be a continuous function on $[0,1]$ satisfying $g(0)=g(1)$. By considering the function $f(x)=g\left(x+\frac{1}{2}\right)-g(x)$ on $\left[0, \frac{1}{2}\right]$, show that $g\left(c+\frac{1}{2}\right)=g(c)$ for some $c \in\left[0, \frac{1}{2}\right]$. Show, more generally, that for any positive integer $n$ there exists a point $c_{n} \in\left[0, \frac{n-1}{n}\right]$ for which $g\left(c_{n}+\frac{1}{n}\right)=g\left(c_{n}\right)$.

  comment

• 1.II.12E

  Analysis

  Part IA, 2005

  State and prove Rolle's Theorem.

  Prove that if the real polynomial $p$ of degree $n$ has all its roots real (though not necessarily distinct), then so does its derivative $p^{\prime}$. Give an example of a cubic polynomial $p$ for which the converse fails.

  comment

• 1.I.3F

  Analysis

  Part IA, 2006

  Let $a_{n} \in \mathbb{R}$ for $n \geqslant 1$. What does it mean to say that the infinite series $\sum_{n} a_{n}$ converges to some value $A$ ? Let $s_{n}=a_{1}+\cdots+a_{n}$ for all $n \geqslant 1$. Show that if $\sum_{n} a_{n}$ converges to some value $A$, then the sequence whose $n$-th term is

  $$\left(s_{1}+\cdots+s_{n}\right) / n$$

  converges to some value $\tilde{A}$ as $n \rightarrow \infty$. Is it always true that $A=\tilde{A}$ ? Give an example where $\left(s_{1}+\cdots+s_{n}\right) / n$ converges but $\sum_{n} a_{n}$ does not.

  comment

• 1.I.4D

  Analysis

  Part IA, 2006

  Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ and $\sum_{n=0}^{\infty} b_{n} z^{n}$ be power series in the complex plane with radii of convergence $R$ and $S$ respectively. Show that if $R \neq S$ then $\sum_{n=0}^{\infty}\left(a_{n}+b_{n}\right) z^{n}$ has radius of convergence $\min (R, S)$. [Any results on absolute convergence that you use should be clearly stated.]

  comment

• 1.II.9E

  Analysis

  Part IA, 2006

  State and prove the Intermediate Value Theorem.

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

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

  and

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

  or otherwise, show that there is a subinterval $\left[a^{\prime}, b^{\prime}\right] \subseteq[a, b]$ such that

  $$\frac{f\left(b^{\prime}\right)-f\left(a^{\prime}\right)}{b^{\prime}-a^{\prime}}=k$$

  Deduce that there exists $c \in(a, b)$ with $f^{\prime}(c)=k$. [You may assume the Mean Value Theorem.]

  comment

• 1.II.10E

  Analysis

  Part IA, 2006

  Prove that if the function $f$ is infinitely differentiable on an interval $(r, s)$ containing $a$, then for any $x \in(r, s)$ and any positive integer $n$ we may expand $f(x)$ in the form

  $$f(a)+(x-a) f^{\prime}(a)+\frac{(x-a)^{2}}{2 !} f^{\prime \prime}(a)+\cdots+\frac{(x-a)^{n}}{n !} f^{(n)}(a)+R_{n}(f, a, x),$$

  where the remainder term $R_{n}(f, a, x)$ should be specified explicitly in terms of $f^{(n+1)}$.

  Let $p(t)$ be a nonzero polynomial in $t$, and let $f$ be the real function defined by

  $$f(x)=p\left(\frac{1}{x}\right) \exp \left(-\frac{1}{x^{2}}\right) \quad(x \neq 0), \quad f(0)=0 .$$

  Show that $f$ is differentiable everywhere and that

  $$f^{\prime}(x)=q\left(\frac{1}{x}\right) \exp \left(-\frac{1}{x^{2}}\right) \quad(x \neq 0), \quad f^{\prime}(0)=0,$$

  where $q(t)=2 t^{3} p(t)-t^{2} p^{\prime}(t)$. Deduce that $f$ is infinitely differentiable, but that there exist arbitrarily small values of $x$ for which the remainder term $R_{n}(f, 0, x)$ in the Taylor expansion of $f$ about 0 does not tend to 0 as $n \rightarrow \infty$.

  comment

• 1.II.11F

  Analysis

  Part IA, 2006

  Consider a sequence $\left(a_{n}\right)_{n \geqslant 1}$ of real numbers. What does it mean to say that $a_{n} \rightarrow$ $a \in \mathbb{R}$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow-\infty$ as $n \rightarrow \infty$ ? Show that for every sequence of real numbers there exists a subsequence which converges to a value in $\mathbb{R} \cup\{\infty,-\infty\}$. [You may use the Bolzano-Weierstrass theorem provided it is clearly stated.]

  Give an example of a bounded sequence $\left(a_{n}\right)_{n \geqslant 1}$ which is not convergent, but for which

  $$a_{n+1}-a_{n} \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$$

  comment

• 1.II.12D

  Analysis

  Part IA, 2006

  Let $f_{1}$ and $f_{2}$ be Riemann integrable functions on $[a, b]$. Show that $f_{1}+f_{2}$ is Riemann integrable.

  Let $f$ be a Riemann integrable function on $[a, b]$ and set $f^{+}(x)=\max (f(x), 0)$. Show that $f^{+}$and $|f|$ are Riemann integrable.

  Let $f$ be a function on $[a, b]$ such that $|f|$ is Riemann integrable. Is it true that $f$ is Riemann integrable? Justify your answer.

  Show that if $f_{1}$ and $f_{2}$ are Riemann integrable on $[a, b]$, then so is $\max \left(f_{1}, f_{2}\right)$. Suppose now $f_{1}, f_{2}, \ldots$ is a sequence of Riemann integrable functions on $[a, b]$ and $f(x)=\sup _{n} f_{n}(x)$; is it true that $f$ is Riemann integrable? Justify your answer.

  comment

• Paper 1, Section I, $3 F$

  Analysis

  Part IA, 2007

  Prove that, for positive real numbers $a$ and $b$,

  $$2 \sqrt{a b} \leqslant a+b$$

  For positive real numbers $a_{1}, a_{2}, \ldots$, prove that the convergence of

  $$\sum_{n=1}^{\infty} a_{n}$$

  implies the convergence of

  $$\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n}$$

  comment

• Paper 1, Section I, D

  Analysis

  Part IA, 2007

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

  Find the value of $R$ for the power series

  $$\sum_{n=1}^{\infty} \frac{z^{n}}{n}$$

  comment

• Paper 1, Section II, F

  Analysis

  Part IA, 2007

  Let $a_{1}=\sqrt{2}$, and consider the sequence of positive real numbers defined by

  $$a_{n+1}=\sqrt{2+\sqrt{a}_{n}}, \quad n=1,2,3, \ldots$$

  Show that $a_{n} \leqslant 2$ for all $n$. Prove that the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

  Suppose instead that $a_{1}=4$. Prove that again the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

  Prove that the limits obtained in the two cases are equal.

  comment

• Paper 1, Section II, E

  Analysis

  Part IA, 2007

  State and prove the Mean Value Theorem.

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that, for every $x \in \mathbb{R}, f^{\prime \prime}(x)$ exists and is non-negative.

  (i) Show that if $x \leqslant y$ then $f^{\prime}(x) \leqslant f^{\prime}(y)$.

  (ii) Let $\lambda \in(0,1)$ and $a<b$. Show that there exist $x$ and $y$ such that

  $$f(\lambda a+(1-\lambda) b)=f(a)+(1-\lambda)(b-a) f^{\prime}(x)=f(b)-\lambda(b-a) f^{\prime}(y)$$

  and that

  $$f(\lambda a+(1-\lambda) b) \leqslant \lambda f(a)+(1-\lambda) f(b) .$$

  comment

• Paper 1, Section II, E

  Analysis

  Part IA, 2007

  Let $a<b$ be real numbers, and let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Show that $f$ is bounded on $[a, b]$, and that there exist $c, d \in[a, b]$ such that for all $x \in[a, b]$, $f(c) \leqslant f(x) \leqslant f(d)$.

  Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that

  $$\lim _{x \rightarrow+\infty} g(x)=\lim _{x \rightarrow-\infty} g(x)=0$$

  Show that $g$ is bounded. Show also that, if $a$ and $c$ are real numbers with $0<c \leqslant g(a)$, then there exists $x \in \mathbb{R}$ with $g(x)=c$.

  comment

• Paper 1, Section II, D

  Analysis

  Part IA, 2007

  Explain carefully what it means to say that a bounded function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

  Prove that every continuous function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

  For each of the following functions from $[0,1]$ to $\mathbb{R}$, determine with proof whether or not it is Riemann integrable:

  (i) the function $f(x)=x \sin \frac{1}{x}$ for $x \neq 0$, with $f(0)=0$;

  (ii) the function $g(x)=\sin \frac{1}{x}$ for $x \neq 0$, with $g(0)=0$.

  comment