Divergence Test

Theorem

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

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

$\blacksquare$

Also known as

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