\displaystyle $\begin{align}
\sum_{n = 0}^{\infty} x^n & = \frac{1}{1 - x} & /\int\limits_0^y dx \\
\sum_{n = 0}^{\infty} \frac{y^{n + 1}}{n + 1} & = -\ln(1 - y) & /\int\limits_0^q dy \\
\sum_{n = 0}^{\infty} \frac{q^{n + 2}}{(n + 1)(n + 2)} & = q + (1 - q) \ln(1 - q) \\
\sum_{n = 1}^{\infty} \frac{q^{n + 1}}{n(n + 1)} & = q + (1 - q) \ln(1 - q) \\
\sum_{n = 1}^{\infty} \frac{q^n}{n(n + 1)} & = 1 + \frac{(1 - q) \ln(1 - q)}{q}
\end{align}$