Degree difference test

Statement

For rational functions

Consider a series of the form:

$\sum _{k=k_0}^{\mathrm{\infty }}\frac{p(k)}{q(k)}$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\ge k_0$. Then the following rules hold:

1. If $\mathrm{deg}q-\mathrm{deg}p\ge 2$, then the series is an absolutely convergent series.
2. If $\mathrm{deg}q-\mathrm{deg}p=1$, then the series diverges, but the terms limit to zero.
3. If $\mathrm{deg}p\ge \mathrm{deg}q$, then the series diverges, and the terms do not limit to zero.

Related tests

• Basic comparison test
• Limit comparison test
• Root test
• Ratio test