Degree difference test

Statement

Unsigned sum version for rational functions

Consider a series of the form:

$\sum _{k=k_{0}}^{\infty }{\frac {p(k)}{q(k)}}$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\geq k_{0}$. Then the following rules hold:

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

Signed sum version for rational functions

Consider a series of the form:

$\sum _{k=k_{0}}^{\infty }{\frac {(-1)^{k}p(k)}{q(k)}}$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\geq k_{0}$. Then the following rules hold:

1. If $\!\deg q-\deg p\geq 2$, then the series is an absolutely convergent series.
2. If $\!\deg q-\deg p=1$, then the series is a conditionally convergent series, i.e., it is convergent but not absolutely convergent.
3. If $\!\deg p\geq \deg q$, then the series diverges.

Version for generalizations of polynomials

The general version replaces polynomials by function obtained as linear combinations of positive power functions, e.g., we could have $p(k)=k^{3/2}+5k^{4/3}-2{\sqrt {k}}$. The degree of such a function is the largest positive power with a nonzero coefficient.

1. If the degree difference (degree of denominator - degree of numerator) is strictly greater than 1, then the series is an absolutely convergent series.
2. If the degree difference (degree of denominator - degree of numerator) is strictly greater than 0 and at most equal to 1, then the series diverges but the terms approach zero.
3. If the degree of numerator is greater than or equal to the degree of denominator, then the series diverges and the terms do not approach zero.

Related tests

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