Divergence Test

Theorem

Let $\sequence {a_n}$ be a sequence in $\R$ which does not converge to $0$.


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

The is also known as the $n$th term test.

The reason for this is neither apparent nor obvious.

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