How do you find the sum of this alternating series? $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)(n+1)!}.$$ 
I found out from my fellow peers at stack exchange see here, that this series converges from the alternating series test. But how do you find the sum? I know if you use wolfram alpha you get: 0.861528, but my question is what steps you use to achieve it?

Here's a familiar trick for summing such series. This is $f(1)$ where
$$f(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)(n+1)!}.$$
Then
$$f'(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(n+1)!}$$
which you can write in closed form. Integrate to get $f(x)$ etc.
