calculus - Sum $sum _{ n=1 }^{ infty }{ frac { 2n }{ left( n+1 right) ! } } $

I need find a Telescopic or Geometric Series but I dont know how do that. I tried everything but nothing work. help me please

$$\sum _{ n=1 }^{ \infty }{ \frac { 2n }{ \left( n+1 \right) ! } } $$

Answer

We have $$\sum_{n = 1}^\infty \frac{2n}{(n+1)!} = 2\sum_{n = 1}^\infty \frac{(n+1) - 1}{(n+1)!} = 2\sum_{n = 1}^\infty \left(\frac{1}{n!} - \frac{1}{(n+1)!}\right)$$