Definition:Condensed Series

Definition

Let $\sequence {a_n}: n \mapsto \map a n$ 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 $\sequence {a_n}$: $a_n > 0, a_{n + 1} \le a_n$, and $a_n \to 0$ as $n \to +\infty$.

The series:

$\ds \sum_{n \mathop = 1}^\infty 2^n \map a {2^n}$

is called the condensed form of the series:

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

Also see

By the Cauchy Condensation Test, the non-condensed series converges if and only if the condensed series converges.

Sources

• 2009: Steven G. Krantz: Discrete Mathematics Demystified: $\S 13.10$