# comparison test

The series

 $\sum_{i=0}^{\infty}a_{i}$

with real $a_{i}$ is absolutely convergent if there is a sequence $(b_{n})_{n\in\mathbb{N}}$ with positive real $b_{n}$ such that

 $\sum_{i=0}^{\infty}b_{i}$

is and for all sufficiently large $k$ holds $|a_{k}|\leq b_{k}$.

Also, the series $\sum a_{i}$ is divergent if there is a sequence $(b_{n})$ with positive real $b_{n}$, so that $\sum b_{i}$ is divergent and $a_{k}\geq b_{k}$ for all sufficiently large $k$.

Title comparison test ComparisonTest 2013-03-22 13:21:48 2013-03-22 13:21:48 mathwizard (128) mathwizard (128) 4 mathwizard (128) Theorem msc 40A05