Question

Find whether the following series is convergent or divergent:
$\frac{2}{1^p}+\frac{3}{2^p}+\frac{4}{3^p}+\frac{5}{4^p}+...$

Answer 1

$S_n=\frac{2}{1^p}+\frac{3}{2^p}+\frac{4}{3^p}+\dots =\sum _{n=1}^{\infty }\frac{n+1}{n^p}=\sum _{n=1}^{\infty }\frac{1}{n^{p-1}}+\sum _{n=1}^{\infty }\frac{1}{n^p}$

First term is a p-series which will converge for $p>2$ and diverge for $p\le 2$

Second term is also a p-series which will coverge for $p>1$ and diverge for $p\le 1$

Therefore $S_n$ is convergent for $p>2$ and divergent for $p\le 2$