(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.