Cauchy Condensation Test

Theorem
Let $\left \langle {a_n} \right \rangle$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero.

That is, let $\forall n \in \N: a_n > 0, a_{n+1} \le a_n, a_n \to 0$ as $n \to \infty$.

Then the series:


 * $\displaystyle \sum_{n=1}^\infty a_n$

converges iff the series:


 * $\displaystyle \sum_{n=1}^\infty 2^n a_{2^n}$

converges.