What Is the Sum of the Series \( \sum_{n=1}^\infty \frac{n}{(n+1)!} \)?

[hobomath]

Find the sum of this series:
$$ \sum_{n=1}^\infty \frac{n}{(n+1)!} $$

I'm really struggling with this one.. Any help will be highly appreciated. Thanks you.

 

[mrtwhs]

Look at the sequence of partial sums.

\(\displaystyle S_1 = \dfrac{1}{2}\), \(\displaystyle S_2 = \dfrac{5}{6}\), \(\displaystyle S_3 = \dfrac{23}{24}\), \(\displaystyle S_4 = \dfrac{119}{120}\), \(\displaystyle S_5 = \dfrac{719}{720}\), etc.

Once you guess the obvious answer for \(\displaystyle S_n\), you can prove it by induction. Then take the limit as \(\displaystyle n \rightarrow \infty\)