Degree difference test: Difference between revisions

Revision as of 00:08, 11 April 2012

Statement

Statement for rational functions

The four different scenarios are discussed below:

Integration of unsigned rational function:
$\int_{x_{0}}^{\infty} \frac{p (x)}{q (x)} d x$
where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q (x)$ is nonzero everywhere on $[x_{0}, \infty)$ .
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 $k \geq k_{0}$ .
Integration of trigonometric function times unsigned rational function: Integration of the form
$\int_{x_{0}}^{\infty} \frac{p (x) \sin x}{q (x)} d x$
where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q (x)$ is nonzero everywhere on $[x_{0}, \infty)$ .
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

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 $p (k) = k^{3 / 2} + 5 k^{4 / 3} - 2 \sqrt{k}$ . 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)
$\deg p \geq \deg q$	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

@@ Line 1: / Line 1: @@
-==Statement for unsigned versions==
+==Statement==
-===Integration version for rational functions===
+===Statement for rational functions===
-Consider an integration of the form:
+The four different scenarios are discussed below:
-<math>\int_{x_0}^\infty \frac{p(x)}{q(x)} \, dx</math>
+# '''Integration of unsigned rational function''':<br><math>\int_{x_0}^\infty \frac{p(x)}{q(x)} \, dx</math><br>where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(x)</math> is nonzero everywhere on <math>[x_0,\infty)</math>.
+# '''Series of unsigned rational function''': Series of the form <br><math>\sum_{k=k_0}^\infty \frac{p(k)}{q(k)}</math><br>where <math>p</math> and <matH>q</math> are polynomials, where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>.
+# '''Integration of trigonometric function times unsigned rational function''': Integration of the form <br><math>\int_{x_0}^\infty \frac{p(x) \sin x}{q(x)} \, dx</math><br>where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(x)</math> is nonzero everywhere on <math>[x_0,\infty)</math>.
+# '''Series of signed rational function''': Series of the form <br><math>\sum_{k=k_0}^\infty \frac{(-1)^kp(k)}{q(k)}</math><br>where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>.
-where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(x)</math> is nonzero everywhere on <math>[x_0,\infty)</math>. Then:
+Then the following cases need to be made and conclusions drawn:
-# If <math>\! \deg q - \deg p > 1</math> (which in particular means that it is <matH>\ge 2</math>), the integral converges.
+{| class="sortable" border="1"
-# If <math>\! 0 < \deg q - \deg p \le 1</math> (which in particular means that it is equal to 1), the integral does not converge, even though the integrand approaches zero.
+! 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)
-# If <math>\! \deg p \ge \deg q</math>, then the integral diverges, because the integrand does not approach zero.
+|-
+| <math>\! \deg q - \deg p > 1</math> || <math>\! \deg q - \deg p \ge 2</math>)|| integral converges (absolutely) || summation converges (absolutely) || integral converges (absolutely) || summation converges (absolutely)
+|-
+| <math>\! 0 < \deg q - \deg p \le 1</math> || <math>\! \deg q - \deg p = 1</math> || integral diverges, but integrand approaches zero || summation diverges, but terms approach zero || integral converges (conditionally) || summation converges (conditionally)
+|-
+| <math>\! \deg q - \deg p \le 0</math> || <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
+|}
-===Unsigned sum version for rational functions===
+==Statement for quotients of sums of power functions===
-Consider a series of the form:
+The general version replaces polynomials by function obtained as linear combinations of positive power functions, e.g., we could have <math>p(k) = k^{3/2} + 5k^{4/3} - 2\sqrt{k}</math>. We assume both <math>p</math> and <matH>q</matH> are of this type. For such functions, we define the ''degree'' as the highest exponent with a nonzero coefficient.
-<math>\sum_{k=k_0}^\infty \frac{p(k)}{q(k)}</math>
+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:
-where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>. Then the following rules hold:
+{| 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)
-# If <math>\! \deg q - \deg p > 1</math> (which in particular means that it is <matH>\ge 2</math>), then the series is an [[absolutely convergent series]].
+|-
-# If <math>\! 0 < \deg q - \deg p \le 1</math> (which in particular means that it is equal to 1), then the series diverges, but the terms limit to zero.
+| <math>\! \deg q - \deg p > 1</math> ||  integral converges (absolutely) || summation converges (absolutely) || integral converges (absolutely) || summation converges (absolutely)
-# If <math>\! \deg p \ge \deg q</math>, then the series diverges, and the terms do not limit to zero.
+|-
+| <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)
-===Version for generalizations of polynomials===
+|-
+| <math>\! \deg p \ge \deg q</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
-The general version replaces polynomials by function obtained as linear combinations of positive power functions, e.g., we could have <math>p(k) = k^{3/2} + 5k^{4/3} - 2\sqrt{k}</math>. The degree of such a function is the largest positive power with a nonzero coefficient.
+|}
-# If the degree difference (degree of denominator - degree of numerator) is strictly greater than 1, then the series is an [[absolutely convergent series]].
-# 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.
-# 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:
-<math>\int_{x_0}^\infty \frac{p(x) \sin x}{q(x)} \, dx</math>
-where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(x)</math> is nonzero everywhere on <math>[x_0,\infty)</math>. Then:
-# If <math>\! \deg q - \deg p > 1</math> (which in particular means that it is <matH>\ge 2</math>), the integral converges absolutely.
-# If <math>\! 0 < \deg q - \deg p \le 1</math> (which in particular means that it is equal to 1), the integral converges conditionally but not absolutely.
-# If <math>\! \deg p \ge \deg q</math>, then the integral diverges, because the integrand does not approach zero.
-===Signed sum version for rational functions===
-Consider a series of the form:
-<math>\sum_{k=k_0}^\infty \frac{(-1)^kp(k)}{q(k)}</math>
-where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>. Then the following rules hold:
-# If <math>\! \deg q - \deg p > 1</math> (which in particular means that it is <matH>\ge 2</math>), then the series is an [[absolutely convergent series]].
-# If <math>\! 0 < \deg q - \deg p \le 1</math> (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.
-# If <math>\! \deg p \ge \deg q</math>, then the series diverges.
 ==Related tests==