Can't understand this step in Zeta-function transformations Sorry for exceptional dullness, but I can't understand this step in Zeta-function transformations: $$ζ(s)= \sum_{n=1}^\infty \frac{1}{n^s}= \sum_{n=1}^\infty n (\frac{1}{n^s}-\frac{1}{(n+1)^s})$$

\begin{align*} \sum_{n=1}^{\infty} \frac{n}{(n+1)^{s}} &= \sum_{n=1}^{\infty} \left[ \frac{n+1}{(n+1)^{s}}-\frac{1}{(n+1)^{s}} \right] \\\ &= \sum_{n=2}^{\infty} \frac{1}{n^{s-1}}- \sum_{n=2}^{\infty} \frac{1}{n^{s}} \\\ &= \left( \sum_{n=1}^{\infty} \frac{1}{n^{s-1}} -1 \right)- \left( \sum_{n=1}^{\infty} \frac{1}{n^{s}} -1 \right) \\\ &= \sum_{n=1}^{\infty} \frac{1}{n^{s-1}}- \sum_{n=1}^{\infty} \frac{1}{n^{s}} \end{align*} Hence, \begin{align*} \zeta(z)&=\sum_{n=1}^{\infty}\frac{1}{n^{s}} = \sum_{n=1}^{\infty} \left[\frac{1}{n^{s-1}} - \frac{n}{(n+1)^{s}} \right] = \sum_{n=1}^{\infty} n \left[\frac{1}{n^{s}} - \frac{1}{(n+1)^{s}} \right] \end{align*}