This test involves a lot of assumptions about the series, so it should only be used when sure it would apply.

Given a series $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ and a sufficently large $M$ if all three of the following conditions are met:

1. $a_n$ is positive for all $M\ge n$
2. $a_n$ is decreasing for all $M\ge n$
3. $a_n$ is continuous for all $M\ge n$

The assuming $f(n)=a_n$ convergence of the series can be determined by the improper integral:

• If ${\int }_n^{\mathrm{\infty }}f(n)dn$ diverges, then $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ diverges as well.
• If ${\int }_n^{\mathrm{\infty }}f(n)dn$ converges, then $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ converges as well.