Limit 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

Suppose we have two series of (eventually) positive terms:

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

and

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

Suppose, further, that the limit

$\lim _{k\to \infty }{\frac {a_{k}}{b_{k}}}$

exists and is a nonzero real number. Then, the series $\sum _{k=1}^{\infty }a_{k}$ is a convergent series if and only if the series $\sum _{k=1}^{\infty }b_{k}$ is a convergent series.

Related tests

• Basic comparison test
• Degree difference test
• Root test
• Ratio test