Divergence Test
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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.