Basic comparison test

This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

For series of nonnegative terms

Suppose we have two series of nonnegative terms:

$\sum _{k=1}^{\infty }a_{k}$

and

$\sum _{k=1}^{\infty }b_{k}$

such that there exists a positive integer $k_{0}$ such that:

$a_{k}\leq b_{k}\ \forall k\geq k_{0}$

Then, we have the following:

1. If the series $\sum _{k=1}^{\infty }a_{k}$ diverges, the series $\sum _{k=1}^{\infty }b_{k}$ also diverges.
2. If the series $\sum _{k=1}^{\infty }b_{k}$ converges, the series $\sum _{k=1}^{\infty }a_{k}$ also converges.