Convergent Series $\frac{1}{n^q}, \ \ q>1$

Question

How to show the following result about series? Thank you!

Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$

Answer 1

The sequence of partial sums $s_m=\sum_{n=1}^m \frac{1}{n^q}$ is increasing. So if we can show that the sequence $(s_m)$ is bounded above, then it will follow that the sequence $(s_m)$ converges, that is, the series converges.

But note that a graph of $y=\frac{1}{x^q}$ shows that $$\sum_1^m \frac{1}{n^q}\lt 1+\int_1^{\infty} \frac{1}{x^q}\,dx,\tag{1}$$ and it is not hard to show that the integral on the right of (1) converges.

Remark: The technique we used is called th Integral Test. Alternately, we could use the Cauchy Condensation Test (please see the Wikipedia article).