The whole question looks like-
Prove that, $\sum_{n\ge0}\frac{(-1)^n}{(2n)!+1}$ is convergent. Find its value. (Options: $\pi/2$, $4\pi$, $\pi/4$, $2\pi$)
I have showed the convergence part.
Let, $\sum_{n\ge0}\frac{(-1)^n}{2n!+1}=\sum_{n\ge0}(-1)^n a_n$.
Now, $a_n\ge0\ \forall n\in\Bbb{N}$, $\{a_n\}$ is decreasing sequence and converges to $0$. So, by alternative series test $\sum_{n\ge0}(-1)^n a_n$ is convergent.
But I can't find its value. If there is no factorial part beside $2n$ in the denominator, I can use power series representation of $\arctan(x)$ at $0$ to get its value.
Can anybody solve the problem? Thanks for assistance in advance.