Use induction to prove that for any positive integers $m,s$ $$\sum_{n=1}^{s}\frac{n!}{(n+m)!} = \frac{1}{m-1} \left( \frac{1}{m!} - \frac{(s+1)!}{(s+m)!}\right)$$
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http://math.stackexchange.com/questions/2070606/how-do-i-solve-this-question-based-on-induction – lab bhattacharjee Dec 25 '16 at 11:40
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@lab bhattacharjee I already fixed that problem. The trick is we fix $m$ and induct on $s$. Thank you. – Juniven Acapulco Dec 26 '16 at 00:50