# Limit alternating series

Show that for any $b>0$ and $n>0$, the alternating sum of the series

$$a_k=\prod_{i=1}^k\frac{1+b/(n+i)}{1+b/(n+1)}$$

converges such that

$$\sum_{k=1}^{\infty} (-1)^k a_k \leq -\frac{1}{2}$$

In fact

$$a_k=a_{k-1}\frac{(n+k+b)(n+1)}{(n+k)(n+1+b)}$$

Can anybody help me?

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Have you tried the ratio test for proving convergence? – Timmy Turner Aug 2 '12 at 10:02
There are some indexing problems. Your sum starts with $k=0$, but definition of $a_k$ assumes that $k\geq 1$ – Norbert Aug 2 '12 at 10:04
I know the series converges. The issue is to find the limit of the sum. – Alexander Aug 2 '12 at 10:08
The indexing in the summation is correct now. Thanks for the comment. – Alexander Aug 2 '12 at 10:19