(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) $x=-\frac{1}{4}$.