# 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.