Degree difference test: Difference between revisions
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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 <math>x</math> with exponent infinitesimally greater than 0, writing it as <math>0^+</math>. So the degree of <math>x^2(\ln x)^5</math> is <math>2^+</math>. The subtraction rule for determining degree differences is that <math>a - b^+ = (a - b)^-</math>, whereas <math>a^+ - b = (a - b)^+</math>, and <math>a^+ - b^+</math> is indeterminate (though we may be able to manipulate a bit and avoid that situation. | |||
We now have the following cases: | |||
{| class="sortable" border="1" | |||
! 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) | |||
|- | |||
| <math>\! \deg q - \deg p > 1^+</math> || integral converges (absolutely) || summation converges (absolutely) || integral converges (absolutely) || summation converges (absolutely) | |||
|- | |||
| <math>\! \deg q - \deg p = 1^+</math> || 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 | |||
|- | |||
| <math>\! 0 < \deg q - \deg p \le 1</math> || integral diverges, but integrand approaches zero || summation diverges, but terms approach zero || integral converges (conditionally) || summation converges (conditionally) | |||
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| <math>\! \deg q -\deg p \le 0</math> || 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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The case of degree difference <math>1^+</math> is inconclusive, and needs to be settled using the [[integral test]]. The integral test tells us that if <math>p(x) = 1, q(x) = x \ln(x + 2)</math>, the series diverges, but if <math>p(x) = 1, q(x) = x (\ln(x + 2))^2</math>, 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. | |||
==Related tests== | ==Related tests== | ||
Revision as of 20:50, 30 June 2012
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:
- Integration of rational function:
where and are polynomials, is not the zero polynomial, and is nonzero everywhere on . - Series of (unsigned) rational function: Series of the form
where and are polynomials, where and are polynomials, is not the zero polynomial, and is nonzero for all integers . - Integration of sine function times rational function: Integration of the form
where and are polynomials, is not the zero polynomial, and is nonzero everywhere on . - Series of signed rational function: Series of the form
where and are polynomials, is not the zero polynomial, and is nonzero for all integers .
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) |
---|---|---|---|---|---|
integral converges (absolutely) | summation converges (absolutely) | integral converges (absolutely) | summation converges (absolutely) | ||
integral diverges, but integrand approaches zero | summation diverges, but terms approach zero | integral converges (conditionally) | summation converges (conditionally) | ||
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 . We assume both and 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) |
---|---|---|---|---|
integral converges (absolutely) | summation converges (absolutely) | integral converges (absolutely) | summation converges (absolutely) | |
integral diverges, but integrand approaches zero | summation diverges, but terms approach zero | integral converges (conditionally) | summation converges (conditionally) | |
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 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 with exponent infinitesimally greater than 0, writing it as . So the degree of is . The subtraction rule for determining degree differences is that , whereas , and 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) |
---|---|---|---|---|
integral converges (absolutely) | summation converges (absolutely) | integral converges (absolutely) | summation converges (absolutely) | |
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 | |
integral diverges, but integrand approaches zero | summation diverges, but terms approach zero | integral converges (conditionally) | summation converges (conditionally) | |
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 is inconclusive, and needs to be settled using the integral test. The integral test tells us that if , the series diverges, but if , 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.