Divergence Test

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

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

Proof
Assume that $$\sum_{i=1}^\infty a_n$$ converges.

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

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