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?