Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

summation induction factorial

Solution 1:

Hint: $$1-\frac1{(n+1)!}+\frac{n+1}{(n+2)!} = 1-\frac{n+2}{(n+2)!}+\frac{n+1}{(n+2)!} = 1-\frac{1}{(n+2)!}.$$

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