What is the P Series Test ???

$$\sum\limits_{n=1}^{\infty}\frac{1}{n^p}$$ This series converges, when when $$ p>1 $$ and diverges if $$p\leq 1 $$

Examples

$$ \sum\limits_{n=1}^{\infty}\frac{1}{n^{\color{red}{3}}}, \text{p=3}, $$

Since $$ p=3 $$, this series converges.

$$ \sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}} $$

Since $$ p=\frac{1}{2} $$, this series diverges.

$$ \sum\limits_{n=1}^{\infty}n^{-\frac{3}{2}} , p=\frac{3}{2} , $$

Remeber $$ n^{-\frac{3}{2}} = \frac{1}{n^{\frac{3}{2 } } }$$. Since $$ p=\frac{3}{2}$$, this series converges.

$$ \sum\limits_{n=1}^{\infty}n^{-1} , $$

Remeber $$ n^{-1} = \frac{1}{n^1} $$. Since $$ p= 1 $$ , this series diverges.