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

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.

Therefore it must be the case that $\displaystyle \sum_{i=1}^\infty a_n$ diverges.

$\blacksquare$