Does the series $\sum_{n=1}^\infty \frac{1}{n^p}$ converge for $p\in [1, +\infty)$

Question

Does the series $$\sum_{n=1}^\infty \frac{1}{n^p}$$ converge for $p\in [1, +\infty)$? If it does not, how to prove it?

Answer 1

For $p = 1$, this is the harmonic series which is divergent:

$$1 + \frac12 + \underbrace{\frac13 + \frac14}_{>2\cdot \frac12} + \underbrace{\frac15 + \cdots + \frac 18}_{> 4\cdot \frac18 = \frac12} + \underbrace{\frac19 + \cdots + \frac{1}{16}}_{> 8 \frac{1}{16} = \frac12} + \cdots $$

and from this it's clear that the partial sums of the series are unbounded, i.e. the series diverges.

For $p > 1$, convergence follows most easily from the integral test.

Answer 2

No, the series converges only for $p\in (1,\infty)$, as can be shown quite easily using the integral test. The case $p=1$ yields the harmonic series which is well-known to diverge as can be shows in various ways.

Answer 3

This is a standard Analysis question. It's answered in say Baby Rudin (Principles of Mathematical analysis. You can use the following from Rudin:

If $x_n$ is decreasing and non negative, $\sum x_n$ converges IFF $\sum 2^nx_{2^n}$ converges.