Analysis | Part IA, 2004

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

an+1anbn+1bn\frac{a_{n+1}}{a_{n}} \leqslant \frac{b_{n+1}}{b_{n}}

for all n1n \geqslant 1, and if n=1bn\sum_{n=1}^{\infty} b_{n} converges, then n=1an\sum_{n=1}^{\infty} a_{n} converges.

(ii) Let

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

By considering logcn\log c_{n}, or otherwise, show that cn0c_{n} \rightarrow 0 as nn \rightarrow \infty.

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

(iii) Determine the convergence or otherwise of

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

for (a) x=14x=\frac{1}{4}, (b) x=14x=-\frac{1}{4}.

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