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:
$$ \frac{a+n-1}{\prod_{j=1}^n (a+j)}=\sum_{k=1}^n\frac{b_k}{a+k} $$
$$ b_{k}=\frac{n-k-1}{\prod_{j=1,j\neq k}^n (j-k)}$$
$$ \sum_{n=1}^N u_{n}= \sum_{n=1}^N \sum_{k=1}^n\frac{n-k-1}{(a+k)\prod_{j=1,j\neq k}^n (j-k)}$$
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