Answer
See below.
Work Step by Step
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{p}}=\frac{1}{1^{p}}+\frac{1}{2^{p}}+\frac{1}{3^{p}}+\cdots+\frac{1}{n^{p}}+\cdots\qquad$
which is a p-series
By Example 3 in section 3, the p-series converges for $ p\gt1$ and diverges for $p\leq 1.$
The proof was derived by using the Integral Text.
Examples:
When $p=1$, we have the harmonic series,
$1+\displaystyle \frac{1}{2}+\frac{1}{3}++\frac{1}{n}+...$
which we know from before diverges.
When $p=2$, we have the series:
$1+\displaystyle \frac{1}{4}+\frac{1}{9}++\frac{1}{n^{2}}+$
which converges by the Integral Test.
