Divergence Test

Theorem
Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

If $\displaystyle \lim_{k \to \infty} a_k \ne 0$, then $\displaystyle \sum_{i=1}^\infty a_n$ diverges.

Proof
Seeking a contradiction, assume that $\displaystyle \sum_{i=1}^\infty a_n$ converges.

Then $\displaystyle \lim_{k \to \infty} a_k = 0$ because Terms in Convergent Series Converge to Zero.

By hypothesis, $\displaystyle \lim_{k \to \infty} a_k \ne 0$.

From this contradiction we conclude that $\displaystyle \sum_{i=1}^\infty a_n$ diverges.

Also known as
This theorem is also known as the $n$th Term Test. The reason for this is neither apparent nor obvious.