(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 $0<x<1$.]