Degree difference test

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This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

Statement for rational functions

The four different scenarios are discussed below:

1. Integration of rational function:
   $\int _{x_{0}}^{\infty }{\frac {p(x)}{q(x)}}\,dx$
   where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(x)$ is nonzero everywhere on $[x_{0},\infty )$.
2. Series of (unsigned) rational function: Series of the form
   $\sum _{k=k_{0}}^{\infty }{\frac {p(k)}{q(k)}}$
   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 $k\geq k_{0}$.
3. Integration of sine function times rational function: 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 $[x_{0},\infty )$.
4. Series of signed rational function: 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 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)
$\!\deg q-\deg p>1$	$\!\deg q-\deg p\geq 2$	integral converges (absolutely)	summation converges (absolutely)	integral converges (absolutely)	summation converges (absolutely)
$\!0<\deg q-\deg p\leq 1$	$\!\deg q-\deg p=1$	integral diverges, but integrand approaches zero	summation diverges, but terms approach zero	integral converges (conditionally)	summation converges (conditionally)
$\!\deg q-\deg p\leq 0$	$\!\deg q-\deg p\leq 0$	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

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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)
$\!\deg q-\deg p>1$	integral converges (absolutely)	summation converges (absolutely)	integral converges (absolutely)	summation converges (absolutely)
$\!0<\deg q-\deg p\leq 1$	integral diverges, but integrand approaches zero	summation diverges, but terms approach zero	integral converges (conditionally)	summation converges (conditionally)
$\!\deg q-\deg p\leq 0$	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

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Statement where we allow logarithms and other increasing functions

We can generalize the degree difference test a little bit as follows. We allow the consideration of logarithms, and treat a logarithm as a power of $x$ with exponent infinitesimally greater than 0, writing it as $0^{+}$. So the degree of $x^{2}(\ln x)^{5}$ is $2^{+}$. The subtraction rule for determining degree differences is that $a-b^{+}=(a-b)^{-}$, whereas $a^{+}-b=(a-b)^{+}$, and $a^{+}-b^{+}$ is indeterminate (though we may be able to manipulate a bit and avoid that situation).

We now have the following cases:

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)
$\!\deg q-\deg p>1^{+}$	integral converges (absolutely)	summation converges (absolutely)	integral converges (absolutely)	summation converges (absolutely)
$\!\deg q-\deg p=1^{+}$	inconclusive, but integrand approaches zero	inconclusive, but integrand approaches zero	integral converges, but unclear whether convergence is absolute or conditional	summation converges, but unclear whether convergence is absolute or conditional
$\!0<\deg q-\deg p\leq 1$	integral diverges, but integrand approaches zero	summation diverges, but terms approach zero	integral converges (conditionally)	summation converges (conditionally)
$\!\deg q-\deg p\leq 0$	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

The case of degree difference $1^{+}$ is inconclusive, and needs to be settled using the integral test, possibly combining it with the basic comparison test or limit comparison test. The integral test tells us that if $p(x)=1,q(x)=x\ln(x+2)$, the series diverges, but if $p(x)=1,q(x)=x(\ln(x+2))^{2}$, the series converges. We could formulate a version of the degree difference test that is sensitive to this and avoids use of the integral test, but that may not be worth the effort.

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Related tests

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

Explanation

Explanation for obvious non-convergence

If $\operatorname {deg} (q)-\operatorname {deg} (p)\leq 0$, then none of the series or integrals converges because the summands (respectively, integrands) do not approach zero.

Explanation for absolute convergence

The key thing to remember is that for a polynomial (as well as any of the generalizations discussed) the sign is eventually constant. Further, if $\operatorname {deg} (q)-\operatorname {deg} (p)>0$, then the magnitude of the rational function is eventually decreasing. Multiplying by -1 if necessary, we get a function that is eventually nonnegative, continuous, and decreasing. Thus, we can use the integral test to convert between summations and integrals, and also use a limit comparison test to convert to determining when the following integral converges:

$\int _{x_{0}}^{\infty }{\frac {1}{x^{\operatorname {deg} q-\operatorname {deg} p}}}\,dx$

Let $r=\operatorname {deg} q-\operatorname {deg} p$.

• If $r>1$, the integral becomes $\left[{\frac {x^{1-r}}{1-r}}\right]_{x_{0}}^{\infty }$. The limit at $\infty$ is zero because $1-r$ is negative, so the integral is ${\frac {-x_{0}^{1-r}}{1-r}}={\frac {x_{0}^{1-r}}{r-1}}$. The key point is that this number is finite, hence the summations and integrals converge.
• If $0<r<1$, the integral becomes $\left[{\frac {x^{1-r}}{1-r}}\right]_{x_{0}}^{\infty }$. The limit at $\infty$ is $\infty$, so it does not converge.
• If $r=1$, the integral becomes $[\ln x]_{x_{0}}^{\infty }$. The limit at $\infty$ is $\infty$, so it does not converge.

The upshot is that convergence occurs if and only if the degree difference is greater than 1.

Explanation for conditional convergence

The convergence of the signed summations for $\operatorname {deg} (q)-\operatorname {deg} (p)>0$ essentially follows from the alternating series theorem.

The convergence of the integral involving a trigonometric function follows through a more indirect use of the alternating series theorem. Explicitly, we break up the improper integral as a sum of definite integrals over half-periods of $\sin$. The definite integrals themselves form an alternating series that satisfies the conditions of the alternating series theorem, hence the sum of these converges.

Note that combined with the absolute convergence information, this tells us that $0<\operatorname {deg} (q)-\operatorname {deg} (p)\leq 1$ gives conditional convergence of the signed summation and trigonometric integral and $\operatorname {deg} (q)-\operatorname {deg} (p)>1$ gives absolute convergence of the signed summation and trigonometric integral.