Degree difference test - Revision history

Vipul at 21:54, 22 December 2012

2012-12-22T21:54:50Z

← Older revision		Revision as of 21:54, 22 December 2012
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			{{perspectives}}
	{{convergence test}}		{{convergence test}}
	==Statement==		==Statement==

Vipul at 21:54, 22 December 2012

2012-12-22T21:54:30Z

← Older revision		Revision as of 21:54, 22 December 2012
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	\| <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		\| <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
	\|}		\|}

			<center>{{#widget:YouTube\|id=jnwMdgW4vbw}}</center>

	===Statement for quotients of sums of power functions===		===Statement for quotients of sums of power functions===
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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		\| <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
	\|}		\|}

			<center>{{#widget:YouTube\|id=WiITIPWfAso}}</center>

	===Statement where we allow logarithms and other increasing functions===		===Statement where we allow logarithms and other increasing functions===
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	The case of degree difference <math>1^+</math> 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 <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.		The case of degree difference <math>1^+</math> 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 <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.

			<center>{{#widget:YouTube\|id=GmD4u4YKCZk}}</center>
	==Related tests==		==Related tests==

Vipul: /* Statement where we allow logarithms and other increasing functions */

2012-10-31T00:03:24Z

Statement where we allow logarithms and other increasing functions

← Older revision		Revision as of 00:03, 31 October 2012
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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.		The case of degree difference <math>1^+</math> 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 <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==

Vipul: /* Statement where we allow logarithms and other increasing functions */

2012-10-31T00:02:15Z

Statement where we allow logarithms and other increasing functions

← Older revision		Revision as of 00:02, 31 October 2012
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	===Statement where we allow logarithms and other increasing functions===		===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 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:		We now have the following cases:
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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.		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==

Vipul: /* Explanation for absolute convergence */

2012-07-03T17:04:08Z

Explanation for absolute convergence

← Older revision		Revision as of 17:04, 3 July 2012
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	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 <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>, 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:		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 <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>, 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:

	<math>\int_{x_0}^\infty \frac{1}{x^{\operatorname{deg} q - \operatorname{deg} p} \, dx</math>		<math>\int_{x_0}^\infty \frac{1}{x^{\operatorname{deg} q - \operatorname{deg} p}} \, dx</math>

	Let <math>r = \operatorname{deg} q - \operatorname{deg} p</math>.		Let <math>r = \operatorname{deg} q - \operatorname{deg} p</math>.

Vipul: /* Related tests */

2012-07-03T17:03:31Z

Related tests

← Older revision		Revision as of 17:03, 3 July 2012
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	* [[Root test]]		* [[Root test]]
	* [[Ratio test]]		* [[Ratio test]]

			==Explanation==

			===Explanation for obvious non-convergence===

			If <math>\operatorname{deg}(q) - \operatorname{deg}(p) \le 0</math>, 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 <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>, 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:

			<math>\int_{x_0}^\infty \frac{1}{x^{\operatorname{deg} q - \operatorname{deg} p} \, dx</math>

			Let <math>r = \operatorname{deg} q - \operatorname{deg} p</math>.

			* If <math>r > 1</math>, the integral becomes <math>\left[\frac{x^{1-r}}{1 - r}\right]_{x_0}^\infty</math>. The limit at <math>\infty</math> is zero because <matH>1 - r</math> is negative, so the integral is <math>\frac{-x_0^{1-r}}{1 - r} = \frac{x_0^{1-r}}{r - 1}</math>. The key point is that this number is finite, hence the summations and integrals converge.
			* If <math>0 < r < 1</math>, the integral becomes <math>\left[\frac{x^{1-r}}{1 - r}\right]_{x_0}^\infty</math>. The limit at <math>\infty</math> is <math>\infty</math>, so it does not converge.
			* If <math>r = 1</math>, the integral becomes <math>[\ln x]_{x_0}^\infty</math>. The limit at <math>\infty</math> is <math>\infty</math>, 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 <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math> 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 <math>\sin</math>. 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 <math>0 < \operatorname{deg}(q) - \operatorname{deg}(p) \le 1</math> gives conditional convergence of the signed summation and trigonometric integral and <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 1</math> gives absolute convergence of the signed summation and trigonometric integral.

Vipul at 20:50, 30 June 2012

2012-06-30T20:50:37Z

← Older revision		Revision as of 20:50, 30 June 2012
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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)
			\|-
			\| <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
			\|}

			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==

Vipul at 15:50, 4 May 2012

2012-05-04T15:50:24Z

← Older revision		Revision as of 15:50, 4 May 2012
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			{{convergence test}}
	==Statement==		==Statement==

Vipul: /* Statement for rational functions */

2012-04-11T00:32:39Z

Statement for rational functions

← Older revision		Revision as of 00:32, 11 April 2012
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	# '''Integration of 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>.		# '''Integration of 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>.		# '''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 sine 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>.		# '''Integration of sine function times 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>.		# '''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>.

Vipul: /* Statement for rational functions */

2012-04-11T00:31:00Z

Statement for rational functions

← Older revision		Revision as of 00:31, 11 April 2012
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	The four different scenarios are discussed below:		The four different scenarios are discussed below:

	# '''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>.		# '''Integration of 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>.		# '''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>.		# '''Integration of sine 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>.		# '''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>.