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:

$\sum _{k=1}^{\mathrm{\infty }}{a}_k$

and

$\sum _{k=1}^{\mathrm{\infty }}{b}_k$

Suppose, further, that the limit

$\underset{k\to \mathrm{\infty }}{lim}\frac{a_k}{b_k}$

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

Related tests

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