How do we calculate this sum $\sum_{n=1}^{\infty} \frac{1}{n(n+1)\cdots(n+p)}$?

real-analysis sequences-and-series

I know that this sum $$\sum_{n=1}^{\infty} \frac{1}{n(n+1)\cdots(n+p)}$$ ($p$ fixed) converges which can be easily proved using the ratio criterion, but I couldn't calculate it.

I need help in this part.

Thanks a lot.


Solution 1:

Hint: $$\frac{p}{n(n+1)\cdots(n+p)}=\frac{1}{(n)(n+1)\cdots (n+p-1)}-\frac{1}{(n+1)(n+2)\cdots (n+p)}.$$

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