Series Divergence

Divergence Test

Given $\sum a_n$ If ${\lim }_{n\to \infty }{a}_n\ne 0$ Then, $\sum a_n$ diverges

Proof By Contrapositive

1. Suppose $\sum a_n$ converges
2. Then, by vanishing condition, ${\lim }_{n\to \infty }{a}_n=0$
3. $\square$

Example

Check if ${\sum }_{n=3}^{\infty }n!$ diverges or converges ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }n!=\infty \ne 0$ Thus, $\sum a_n$ diverges