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Category:3D graphics software
Category:Computer-aided design software
Category:User interface techniquesQ:

how to find the value of $\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{(-2)^{n-k}}{(n+1)^2}$

How to find the value of $$\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{(-2)^{n-k}}{(n+1)^2}$$
I have tried taking complex integral but with no success.

A:

You should use the generating function
$$\sum_{k=0}^\infty (-2)^k x^k=\frac{1}{1+x}$$
This gives, after simplification,
$$\int_0^1\frac{1}{(1+x)^2}\,dx$$
You should be able to solve this from here.

Q:

In a trial-error-correct cycle, are steps to find a solution guaranteed to provide a solution?

In debugging a low level application, I frequently find myself in a situation like this.