<s>[INST] Summation of general series One of the problems in Donald Knuth's Art of Programming is phrased as follows:

Find and prove a simple formula for the sum $$\sum\limits_{n=0}^k\frac{(-1)^n(2n+1)^3}{(2n+1)^4+4}.$$

I have very little experience with summations. My method was to try to find some function such that
$$\frac{(-1)^n(2n+1)^3}{(2n+1)^4+4}=f(x)-f(x-1).$$
However, I have no idea how to do that systematically. Fortunately, partial fraction decomposition happens to give a function in that form, with
$$f(x)=\frac{(-1)^x(x+1)}{4(x+1)^2+1}$$
which is the solution in the back of the book. I was wondering how you might go about finding this solution. This specific problem is volume 1, chapter 1.2.1, problem 11.
 [/INST] By computing the residues of $\frac{x^3}{x^4+4}$ in $x=\pm 1\pm i$ we have:
$$\frac{x^3}{x^4+4}=\frac{1}{2}\left(\frac{x-1}{(x-1)^2+1}+\frac{x+1}{(x+1)^2+1}\right)\tag{1}$$
hence:
$$\begin{eqnarray*}\sum_{n=0}^{k}\frac{(-1)^n(2n+1)^3}{(2n+1)^4+4}&=&\frac{1}{2}\sum_{n=0}^{k}(-1)^n\left(\frac{2n}{4n^2+1}+\frac{2(n+1)}{4(n+1)^2+1}\right)\\&=&\frac{(-1)^k\cdot(k+1)}{4(k+1)^2+1}.\end{eqnarray*}$$
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