Just like you can (sometimes) compute integrals by anti-differentiation, you can sometimes compute infinite series by "anti-difference-ing": if you can find a sequence \((b_n)_{n\ge0}\) such that \(b_{n+1} - b_n = a_n\) for all \(n\), then \(\sum_{n\ge0} a_n = \lim_{n\to\infty} (b_{n+1} - b_0) = -b_0 + \lim_{n\to\infty} b_n\).