Degree difference test

Statement for unsigned versions

Integration version for rational functions

Consider an integration of the form:

$\int _{x_{0}}^{\infty }{\frac {p(x)}{q(x)}}\,dx$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(x)$ is nonzero everywhere on $[x_{0},\infty )$. Then:

1. If $\!\deg q-\deg p>1$ (which in particular means that it is $\geq 2$), the integral converges.
2. If $\!0<\deg q-\deg p\leq 1$ (which in particular means that it is equal to 1), the integral does not converge, even though the integrand approaches zero.
3. If $\!\deg p\geq \deg q$, then the integral diverges, because the integrand does not approach zero.

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 $\!\deg q-\deg p>1$ (which in particular means that it is $\geq 2$), then the series is an absolutely convergent series.
2. If $\!0<\deg q-\deg p\leq 1$ (which in particular means that it is equal to 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.

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.

Statement for signed versions

Integration version for product of sine or cosine with rational function

Consider an integration of the form:

$\int _{x_{0}}^{\infty }{\frac {p(x)\sin x}{q(x)}}\,dx$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(x)$ is nonzero everywhere on $[x_{0},\infty )$. Then:

1. If $\!\deg q-\deg p>1$ (which in particular means that it is $\geq 2$), the integral converges absolutely.
2. If $\!0<\deg q-\deg p\leq 1$ (which in particular means that it is equal to 1), the integral converges conditionally but not absolutely.
3. If $\!\deg p\geq \deg q$, then the integral diverges, because the integrand does not approach 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 $\!\deg q-\deg p>1$ (which in particular means that it is $\geq 2$), then the series is an absolutely convergent series.
2. If $\!0<\deg q-\deg p\leq 1$ (which in particular means that it is equal to 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.

Related tests

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