Definition:Condensed Series

Definition
Let $\left \langle {a_n} \right \rangle: n \mapsto a\left({n}\right)$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero.

That is, for every $n$ in the domain of $\left \langle {a_n} \right \rangle$: $a_n > 0, a_{n+1} \le a_n$, and $a_n \to 0$ as $n \to +\infty$.

The series:


 * $\displaystyle \sum_{n \mathop = 1}^\infty 2^n a\left({2^n}\right)$

is called the condensed form of the series:


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

Also see
By the Cauchy Condensation Test, the non-condensed series converges iff the condensed series converges.