Understand $s \sum_{n=1}^{\infty} n \int_{n}^{n+1} \frac{1}{x^{s+1}} \,dx = s \sum_{n=1}^{\infty} n [\frac{1}{n^{s}} - \frac{1}{(n+1)^{s}}]$

Question

Shouldn't it be the other way around? When evaluating the limit, shouldn't it be

$$s \sum_{n=1}^{\infty} n [\frac{1}{(n+1)^{s}} - \frac{1}{(n)^{s}}]$$? Am I missing something? Thanks for the help! :)

Answer 1

The antiderivative of $x^{-(s+1)}$ is $$ -\frac{1}{s} x^{-s} + C . $$ It's the minus sign that (effectively) reverses the order.