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:

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

and

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

such that there exists a positive integer $k_0$ such that:

$a_k\le b_k\mathrm{\forall }k\ge k_0$

Then, we have the following:

1. If the series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ diverges, the series $\sum _{k=1}^{\mathrm{\infty }}{b}_k$ also diverges.
2. If the series $\sum _{k=1}^{\mathrm{\infty }}{b}_k$ converges, the series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ also converges.