Degree difference test

Statement

Statement for rational functions

The four different scenarios are discussed below:

1. Integration of rational function:
   $\underset{x_0}{\overset{\mathrm{\infty }}{\int }}\frac{p(x)}{q(x)}dx$
   where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(x)$ is nonzero everywhere on $[x_0,\mathrm{\infty })$.
2. Series of (unsigned) rational function: Series of the form
   $\sum _{k=k_0}^{\mathrm{\infty }}\frac{p(k)}{q(k)}$
   where $p$ and $q$ are polynomials, where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\ge k_0$.
3. Integration of sine function times rational function: Integration of the form
   $\underset{x_0}{\overset{\mathrm{\infty }}{\int }}\frac{p(x)\sin x}{q(x)}dx$
   where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(x)$ is nonzero everywhere on $[x_0,\mathrm{\infty })$.
4. Series of signed rational function: Series of the form
   $\sum _{k=k_0}^{\mathrm{\infty }}\frac{(-1{)}^{k}p(k)}{q(k)}$
   where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\ge k_0$.

Then the following cases need to be made and conclusions drawn:

Case on degree difference	Alternative formulation of case using the fact that degrees are integers	Conclusion for improper integral in (1)	Conclusion for series summation in (2)	Conclusion for improper integral in (3)	Conclusion for series summation in (4)
$\mathrm{deg}q-\mathrm{deg}p>1$	$\mathrm{deg}q-\mathrm{deg}p\ge 2$	integral converges (absolutely)	summation converges (absolutely)	integral converges (absolutely)	summation converges (absolutely)
$0<\mathrm{deg}q-\mathrm{deg}p\le 1$	$\mathrm{deg}q-\mathrm{deg}p=1$	integral diverges, but integrand approaches zero	summation diverges, but terms approach zero	integral converges (conditionally)	summation converges (conditionally)
$\mathrm{deg}q-\mathrm{deg}p\le 0$	$\mathrm{deg}q-\mathrm{deg}p\le 0$	integral diverges, integrand does not approach zero	summation diverges, terms do not approach zero	integral diverges, integrand does not approach zero	summation diverges, terms do not approach zero

Statement for quotients of sums of power functions

The general version replaces polynomials by function obtained as linear combinations of positive power functions, e.g., we could have $p(k)=k^{3/2}+5k^{4/3}-2\sqrt{k}$. We assume both $p$ and $q$ are of this type. For such functions, we define the degree as the highest exponent with a nonzero coefficient.

Note that the degree difference test is identical, but we no longer have the alternative formulation because the degrees are no longer guaranteed to be integers. The shortened table is below:

Case on degree difference	Conclusion for improper integral in (1)	Conclusion for series summation in (2)	Conclusion for improper integral in (3)	Conclusion for series summation in (4)
$\mathrm{deg}q-\mathrm{deg}p>1$	integral converges (absolutely)	summation converges (absolutely)	integral converges (absolutely)	summation converges (absolutely)
$0<\mathrm{deg}q-\mathrm{deg}p\le 1$	integral diverges, but integrand approaches zero	summation diverges, but terms approach zero	integral converges (conditionally)	summation converges (conditionally)
$\mathrm{deg}q-\mathrm{deg}p\le 0$	integral diverges, integrand does not approach zero	summation diverges, terms do not approach zero	integral diverges, integrand does not approach zero	summation diverges, terms do not approach zero

Related tests

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