# 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

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

• Hint: We have $2n=(2n+2)-2$. – André Nicolas Feb 5 '15 at 3:56

We have $$\sum_{n = 1}^\infty \frac{2n}{(n+1)!} = 2\sum_{n = 1}^\infty \frac{(n+1) - 1}{(n+1)!} = 2\sum_{n = 1}^\infty \left(\frac{1}{n!} - \frac{1}{(n+1)!}\right)$$
Hint. Since you are looking for a telescoping sum, try something like $$\frac{2n}{(n+1)!}=\frac{a}{n!}-\frac{a}{(n+1)!}\ .$$ Can you find a value of $a$ which makes this work?