Alternating series theorem - Revision history

Vipul at 21:48, 22 December 2012

2012-12-22T21:48:28Z

← Older revision		Revision as of 21:48, 22 December 2012
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			{{perspectives}}
	{{convergence test}}		{{convergence test}}
	==Statement==		==Statement==
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	* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically increasing sequence bounded from above, hence has a limit.		* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically increasing sequence bounded from above, hence has a limit.
	* Both the limits are equal.		* Both the limits are equal.

			<center>{{#widget:YouTube\|id=5YerX2ktuD4}}</center>

	===Concrete version assuming first term is negative===		===Concrete version assuming first term is negative===
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	* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically decreasing sequence bounded from below, hence has a limit.		* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically decreasing sequence bounded from below, hence has a limit.
	* Both the limits are equal.		* Both the limits are equal.
	~~<center>{{#widget:YouTube\|id=5YerX2ktuD4}}</center>~~

	===Eventually variation===		===Eventually variation===

Vipul: /* Related facts */

2012-12-22T21:45:57Z

Related facts

← Older revision		Revision as of 21:45, 22 December 2012
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	* [[Alternating series theorem fails if terms are not monotonically decreasing in magnitude]]: If we drop the condition that the magnitude of the terms is monotonically decreasing, then the series need not converge. Note that it may still ''happen'' to converge, but there is no guarantee by the theorem.		* [[Alternating series theorem fails if terms are not monotonically decreasing in magnitude]]: If we drop the condition that the magnitude of the terms is monotonically decreasing, then the series need not converge. Note that it may still ''happen'' to converge, but there is no guarantee by the theorem.
	* [[Analogue of alternating series theorem if magnitude of terms approaches a positive number]]: The alternating series theorem fails if the magnitude of terms does not approach zero. However, we can formulate an analogous theorem which says that the difference between the limit superior and limit inferior of the partial sums is precisely that positive number limit.		* [[Analogue of alternating series theorem if magnitude of terms approaches a positive number]]: The alternating series theorem fails if the magnitude of terms does not approach zero. However, we can formulate an analogous theorem which says that the difference between the limit superior and limit inferior of the partial sums is precisely that positive number limit.

			<center>{{#widget:YouTube\|id=2FrM8MsIs4U}}</center>

Vipul: /* Eventually variation */

2012-12-22T21:45:25Z

Eventually variation

← Older revision		Revision as of 21:45, 22 December 2012
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	* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically decreasing sequence bounded from below, hence has a limit.		* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically decreasing sequence bounded from below, hence has a limit.
	* Both the limits are equal.		* Both the limits are equal.
			<center>{{#widget:YouTube\|id=5YerX2ktuD4}}</center>

	===Eventually variation===		===Eventually variation===

	The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.		The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.

	~~<center>{{#widget:YouTube\|id=5YerX2ktuD4}}</center>~~

	==Significance==		==Significance==

Vipul: /* Statement */

2012-12-22T21:45:12Z

Statement

← Older revision		Revision as of 21:45, 22 December 2012
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	The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.		The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.

			<center>{{#widget:YouTube\|id=5YerX2ktuD4}}</center>

	==Significance==		==Significance==

Vipul: /* Basic statement */

2012-10-31T22:11:27Z

Basic statement

← Older revision		Revision as of 22:11, 31 October 2012
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	Suppose the following three conditions hold for the series:		Suppose the following three conditions hold for the series:

	# ''Alternating signs'': All the <math>a_k</math>s are nonzero and the sign of <math>a_{k+1}</math> is opposite the sign of <math>a_k</math>.		# ''Alternating signs'': All the <math>a_k</math>s are nonzero and the sign of <math>a_{k+1}</math> is opposite the sign of <math>a_k</math> for all <math>k</math>.
	# ''Monotonically decreasing in magnitude'': <math>\|a_k\| \ge \|a_{k+1}\|</math> for all <math>k</math>.		# ''Monotonically decreasing (i.e., non-increasing) in magnitude'': <math>\|a_k\| \ge \|a_{k+1}\|</math> for all <math>k</math>.
	# ''Terms approach zero'': <math>\lim_{k \to \infty} a_k = 0</math>.		# ''Terms approach zero'': <math>\lim_{k \to \infty} a_k = 0</math>. This is equivalent to saying that <math>\lim_{k \to \infty} \|a_k\| = 0</math>

	Then the series is a [[convergent series]]. It may be an [[absolutely convergent series]] or a [[conditionally convergent series]], depending on whether the series of the absolute values of its terms converges.		Then the series is a [[convergent series]]. It may be an [[absolutely convergent series]] or a [[conditionally convergent series]], depending on whether the series of the absolute values of its terms converges.

Vipul: /* Significance */

2012-07-03T16:24:22Z

Significance

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	* The signed version of the [[degree difference test]] asks for conditions for a rational function <math>\sum (-1)^k \frac{p(k)}{q(k)}</math> to converge. Based on the alternating series theorem, we obtain that convergence occurs if and only if <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>. Moreover, the convergence is absolute if the degree difference is greater than 1 and conditional if the degree difference is greater than 0 and less than or equal to 1.		* The signed version of the [[degree difference test]] asks for conditions for a rational function <math>\sum (-1)^k \frac{p(k)}{q(k)}</math> to converge. Based on the alternating series theorem, we obtain that convergence occurs if and only if <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>. Moreover, the convergence is absolute if the degree difference is greater than 1 and conditional if the degree difference is greater than 0 and less than or equal to 1.
	* The alternating series theorem plays a key role, either directly or via the degree difference test, in the [[rules for determining interval of convergence]]. Specifically, it helps determine for certain [[power series]] whether endpoints are included in the interval of convergence.		* The alternating series theorem plays a key role, either directly or via the degree difference test, in the [[rules for determining interval of convergence]]. Specifically, it helps determine for certain [[power series]] whether endpoints are included in the interval of convergence.

	===Significance for approximate computation of sums of series===		===Significance for approximate computation of sums of series===

			The concrete version of the alternating series theorem can be used to compute upper and lower bounds for the sums of alternating series. Specifically, all partial sums ending at negative terms give lower bounds and all partial sums ending at positive terms give upper bounds.

			In general, alternating series that are absolutely convergent have fairly rapid convergence, and this gives a very effective method for approximating the sum. Alternating series that are ''not'' absolutely convergent have fairly slow convergence, and the method here can be quite slow for approximating the sum (i.e., we need to add a lot of terms to achieve a desired level of accuracy).

	==Related facts==		==Related facts==

Vipul at 16:21, 3 July 2012

2012-07-03T16:21:45Z

← Older revision		Revision as of 16:21, 3 July 2012
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	The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.		The alternating series theorem can be generalized as follows: we do ''not'' require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series ''eventually'' be alternating and ''eventually'' the terms be monotonically decreasing in magnitude.

			==Significance==

			===Significance for showing convergence===

			The alternating series theorem is widely used in showing the convergence of series. Specifically, it helps show the convergence of series of the form <math>\sum (-1)^kb_k</math> where <math>b_k</math> (eventually) have constant sign and are monotonically decreasing in magnitude. A couple of applications are below:

			* The signed version of the [[degree difference test]] asks for conditions for a rational function <math>\sum (-1)^k \frac{p(k)}{q(k)}</math> to converge. Based on the alternating series theorem, we obtain that convergence occurs if and only if <math>\operatorname{deg}(q) - \operatorname{deg}(p) > 0</math>. Moreover, the convergence is absolute if the degree difference is greater than 1 and conditional if the degree difference is greater than 0 and less than or equal to 1.
			* The alternating series theorem plays a key role, either directly or via the degree difference test, in the [[rules for determining interval of convergence]]. Specifically, it helps determine for certain [[power series]] whether endpoints are included in the interval of convergence.
			===Significance for approximate computation of sums of series===

	==Related facts==		==Related facts==

Vipul: /* Statement */

2012-07-03T16:15:53Z

Statement

← Older revision		Revision as of 16:15, 3 July 2012
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	===Concrete version assuming first term is positive===		===Concrete version assuming first term is positive===

	Suppose the ~~above~~ holds and we additionally assume that <math>a_1</math> is positive. ~~(We could formulate a completely analogous statement for <math>a_1</math> negative).~~ Then, <math>a_k</math> is positive for odd <math>k</math> and <math>a_k</math> is negative for even <math>k</math>. The alternating series theorem states, concretely, that:		Suppose the basic statement holds and we additionally assume that <math>a_1</math> is positive. Then, <math>a_k</math> is positive for odd <math>k</math> and <math>a_k</math> is negative for even <math>k</math>. The alternating series theorem states, concretely, that:

	* The partial sums for the first <math>k</math> terms for <math>k</math> odd form a monotonically decreasing sequence bounded from below, hence has a limit.		* The partial sums for the first <math>k</math> terms for <math>k</math> odd form a monotonically decreasing sequence bounded from below, hence has a limit.
	* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically increasing sequence bounded from above, hence has a limit.		* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically increasing sequence bounded from above, hence has a limit.
			* Both the limits are equal.

			===Concrete version assuming first term is negative===

			Suppose the basic statement holds and we additionally assume that <math>a_1</math> is positive. Then, <math>a_k</math> is negative for odd <math>k</math> and <math>a_k</math> is positive for even <math>k</math>. The alternating series theorem states, concretely, that:

			* The partial sums for the first <math>k</math> terms for <math>k</math> odd form a monotonically increasing sequence bounded from above, hence has a limit.
			* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically decreasing sequence bounded from below, hence has a limit.
	* Both the limits are equal.		* Both the limits are equal.
	===Eventually variation===		===Eventually variation===

Vipul at 16:13, 3 July 2012

2012-07-03T16:13:50Z

← Older revision		Revision as of 16:13, 3 July 2012
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	Then the series is a [[convergent series]]. It may be an [[absolutely convergent series]] or a [[conditionally convergent series]], depending on whether the series of the absolute values of its terms converges.		Then the series is a [[convergent series]]. It may be an [[absolutely convergent series]] or a [[conditionally convergent series]], depending on whether the series of the absolute values of its terms converges.

			===Concrete version assuming first term is positive===

			Suppose the above holds and we additionally assume that <math>a_1</math> is positive. (We could formulate a completely analogous statement for <math>a_1</math> negative). Then, <math>a_k</math> is positive for odd <math>k</math> and <math>a_k</math> is negative for even <math>k</math>. The alternating series theorem states, concretely, that:

			* The partial sums for the first <math>k</math> terms for <math>k</math> odd form a monotonically decreasing sequence bounded from below, hence has a limit.
			* The partial sums for the first <math>k</math> terms for <math>k</math> even form a monotonically increasing sequence bounded from above, hence has a limit.
			* Both the limits are equal.
	===Eventually variation===		===Eventually variation===

Vipul: /* Significance of each condition */

2012-07-03T16:03:24Z

Significance of each condition

← Older revision		Revision as of 16:03, 3 July 2012
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	* [[Alternating series theorem fails if signs are not strictly alternating]]: If we replace the strict alternation condition on signs by simply saying that the sign switches between positive and negative infinitely often, that is ''not'' sufficient to guarantee convergence of the alternating series. Note that it may still ''happen'' that the series converges, but there is no guarantee by the theorem.		* [[Alternating series theorem fails if signs are not strictly alternating]]: If we replace the strict alternation condition on signs by simply saying that the sign switches between positive and negative infinitely often, that is ''not'' sufficient to guarantee convergence of the alternating series. Note that it may still ''happen'' that the series converges, but there is no guarantee by the theorem.
	* [[Alternating series theorem fails if terms are not monotonically decreasing in magnitude]]: If we drop the condition that the magnitude of the terms is monotonically decreasing, then the series need not converge. Note that it may still ''happen'' to converge, but there is no guarantee by the theorem.		* [[Alternating series theorem fails if terms are not monotonically decreasing in magnitude]]: If we drop the condition that the magnitude of the terms is monotonically decreasing, then the series need not converge. Note that it may still ''happen'' to converge, but there is no guarantee by the theorem.
	* [[~~Analogoue~~ of alternating series theorem if magnitude of terms approaches a positive number]]: The alternating series theorem fails if the magnitude of terms does not approach zero. However, we can formulate an analogous theorem which says that the difference between the limit superior and limit inferior of the partial sums is precisely that positive number limit.		* [[Analogue of alternating series theorem if magnitude of terms approaches a positive number]]: The alternating series theorem fails if the magnitude of terms does not approach zero. However, we can formulate an analogous theorem which says that the difference between the limit superior and limit inferior of the partial sums is precisely that positive number limit.