I am trying to sum the series
$ \sum u_{n}$
where $ u_{n}=\frac{a+n-1}{\prod_{j=1}^n (a+j)}$ $ a>0$ We have:
$ \frac{a+n-1}{\prod_{j=1}^n (a+j)}=\sum_{k=1}^n\frac{b_k}{a+k} $
$ b_{k}=\frac{n-k-1}{\prod_{j=1,j\neq k}^n (j-k)}$
$ \sum_{n=1}^N u_{n}= \sum_{n=1}^N \sum_{k=1}^n\frac{n-k-1}{(a+k)\prod_{j=1,j\neq k}^n (j-k)}$
...