Definition # Examples **Example**: We have $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1 \quad (*)$The partial sums are$\begin{align} \sum_{n=1}^{\infty} \frac{1}{n(n+1)} &= \sum_{n = 1}^{k} \frac{1}{n} - \frac{1}{n+1} \\ &= \sum_{n=1}^{k} \frac{1}{n} - \sum_{n=1}^{k} \frac{1}{n+1} \\ &= \sum_{n=1}^{k} \frac{1}{n} - \sum_{u= 2}^{k+1} \frac{1}{u} \\ &= \frac{k}{k+1} = \frac{1}{1+\left( \frac{1}{k} \right)} \end{align}$so we can show (\*), by limit of partial sums as $k \to \infty$.