Divergence Test

Theorem
Let $\sequence {a_n}$ be a sequence in $\R$.

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

Proof
We know that Terms in Convergent Series Converge to Zero.

This is the contrapositive statement of this theorem.

Thus, the theorem holds by Rule of Transposition.

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