|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Calculus |Series}}
| | Hi there. My name car warranty is Garland even though it is not the name on my birth certification. What she enjoys performing is to car [http://vendorportal.citypower.co.za/Activity-Feed/My-Profile/UserId/18066 auto warranty] perform croquet but [http://auto.howstuffworks.com/buying-selling/cg-extended-warranty-tips.htm extended auto] warranty she [http://www.gmcertified.com/used-car-warranty hasn't produced] a dime with it. Years in the past we moved to Arizona but my spouse desires us to move. Interviewing is how he [http://www.continentalwarranty.org supports] his family members but his marketing by no means arrives.<br><br>Here is my web-site ... extended car [http://c-lug.com/Activity-Feed/My-Profile/UserId/249456 auto warranty] ([http://Frag5.com/index.php?mod=users&action=view&id=13770 Click at Frag5.com]) |
| | |
| In [[mathematics]], '''Abel's test''' (also known as '''Abel's criterion''') is a method of testing for the [[Convergent series|convergence]] of an [[series (mathematics)|infinite series]]. The test is named after mathematician [[Niels Henrik Abel|Niels Abel]]. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with [[power series]] in [[complex analysis]]. '''Abel's uniform convergence test''' is a criterion for the [[uniform convergence]] of a [[Series (mathematics)|series]] of [[function (mathematics)|functions]] dependent on [[parameter#Mathematical analysis|parameters]].
| |
| | |
| == Abel's test in real analysis ==
| |
| | |
| Suppose the following statements are true:
| |
| | |
| # <math>\sum a_n </math> is a convergent series,
| |
| # {''b''<sub>''n''</sub>} is a monotone sequence, and
| |
| # {''b''<sub>''n''</sub>} is bounded.
| |
| | |
| Then <math>\sum a_nb_n </math> is also convergent.
| |
| | |
| == Abel's test in complex analysis ==
| |
| A closely related convergence test, also known as '''Abel's test''', can often be used to establish the convergence of a [[power series]] on the boundary of its [[radius of convergence|circle of convergence]]. Specifically, Abel's test states that if
| |
| | |
| :<math>
| |
| \lim_{n\rightarrow\infty} a_n = 0\,
| |
| </math>
| |
| | |
| and the series
| |
| | |
| :<math>
| |
| f(z) = \sum_{n=0}^\infty a_nz^n\,
| |
| </math>
| |
| | |
| converges when |''z''| < 1 and diverges when |''z''| > 1, and the coefficients {''a''<sub>''n''</sub>} are ''positive real numbers'' decreasing monotonically toward the limit zero for ''n'' > ''m'' (for large enough ''n'', in other words), then the power series for ''f''(''z'') converges everywhere on the [[unit circle]], except when ''z'' = 1. Abel's test cannot be applied when ''z'' = 1, so convergence at that single point must be investigated separately. Notice that Abel's test can also be applied to a power series with radius of convergence ''R'' ≠ 1 by a simple change of variables ''ζ'' = ''z''/''R''.<ref>(Moretti, 1964, p. 91)</ref>
| |
| | |
| '''Proof of Abel's test:''' Suppose that ''z'' is a point on the unit circle, ''z'' ≠ 1. Then
| |
| | |
| :<math>
| |
| z = e^{i\theta} \quad\Rightarrow\quad z^{\frac{1}{2}} - z^{-\frac{1}{2}} =
| |
| 2i\sin{\textstyle \frac{\theta}{2}} \ne 0
| |
| </math>
| |
| | |
| so that, for any two positive integers ''p'' > ''q'' > ''m'', we can write
| |
| | |
| :<math>
| |
| \begin{align}
| |
| 2i\sin{\textstyle \frac{\theta}{2}}\left(S_p - S_q\right) & =
| |
| \sum_{n=q+1}^p a_n \left(z^{n+\frac{1}{2}} - z^{n-\frac{1}{2}}\right)\\
| |
| & = \left[\sum_{n=q+2}^p \left(a_{n-1} - a_n\right) z^{n-\frac{1}{2}}\right] -
| |
| a_{q+1}z^{q+\frac{1}{2}} + a_pz^{p+\frac{1}{2}}\,
| |
| \end{align}
| |
| </math>
| |
| | |
| where ''S''<sub>''p''</sub> and ''S''<sub>''q''</sub> are partial sums:
| |
| | |
| :<math>
| |
| S_p = \sum_{n=0}^p a_nz^n.\,
| |
| </math>
| |
| | |
| But now, since |''z''| = 1 and the ''a''<sub>''n''</sub> are monotonically decreasing positive real numbers when ''n'' > ''m'', we can also write
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \left| 2i\sin{\textstyle \frac{\theta}{2}}\left(S_p - S_q\right)\right| & =
| |
| \left| \sum_{n=q+1}^p a_n \left(z^{n+\frac{1}{2}} - z^{n-\frac{1}{2}}\right)\right| \\
| |
| & \le \left[\sum_{n=q+2}^p \left| \left(a_{n-1} - a_n\right) z^{n-\frac{1}{2}}\right|\right] +
| |
| \left| a_{q+1}z^{q+\frac{1}{2}}\right| + \left| a_pz^{p+\frac{1}{2}}\right| \\
| |
| & = \left[\sum_{n=q+2}^p \left(a_{n-1} - a_n\right)\right] +a_{q+1} + a_p \\
| |
| & = a_{q+1} - a_p + a_{q+1} + a_p = 2a_{q+1}.\,
| |
| \end{align}
| |
| </math>
| |
| | |
| Now we can apply [[cauchy's convergence test|Cauchy's criterion]] to conclude that the power series for ''f''(''z'') converges at the chosen point ''z'' ≠ 1, because sin(½''θ'') ≠ 0 is a fixed quantity, and ''a''<sub>''q''+1</sub> can be made smaller than any given ''ε'' > 0 by choosing a large enough ''q''.
| |
| | |
| == Abel's uniform convergence test ==
| |
| Abel's uniform convergence test is a criterion for the [[uniform convergence]] of a series of functions or an [[improper integral|improper integration]] of functions dependent on [[parameter#Mathematical analysis|parameters]]. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of [[summation by parts]].
| |
| | |
| The test is as follows. Let {''g''<sub>''n''</sub>} be a [[uniformly bounded]] sequence of real-valued [[continuous function]]s on a set ''E'' such that ''g''<sub>''n''+1</sub>(''x'') ≤ ''g''<sub>''n''</sub>(''x'') for all ''x'' ∈ ''E'' and positive integers ''n'', and let {''ƒ''<sub>''n''</sub>} be a sequence of real-valued functions such that the series Σ''ƒ''<sub>''n''</sub>(''x'') converges uniformly on ''E''. Then Σ''ƒ''<sub>''n''</sub>(''x'')''g''<sub>''n''</sub>(''x'') converges uniformly on ''E''.
| |
| | |
| == Notes ==
| |
| <references/>
| |
| | |
| == References ==
| |
| *Gino Moretti, ''Functions of a Complex Variable'', Prentice-Hall, Inc., 1964
| |
| *{{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Mathematical analysis | publisher=[[Addison-Wesley]] | edition=2nd | isbn=978-0-201-00288-1 | year=1974}}
| |
| *{{mathworld|title=Abel's uniform convergence test|urlname=AbelsUniformConvergenceTest}}
| |
| | |
| == External links ==
| |
| * [http://planetmath.org/encyclopedia/ProofOfAbelsTestForConvergence.html Proof (for real series) at PlanetMath.org]
| |
| | |
| [[Category:Mathematical series]]
| |
| [[Category:Convergence tests]]
| |
| [[Category:Articles containing proofs]]
| |
Hi there. My name car warranty is Garland even though it is not the name on my birth certification. What she enjoys performing is to car auto warranty perform croquet but extended auto warranty she hasn't produced a dime with it. Years in the past we moved to Arizona but my spouse desires us to move. Interviewing is how he supports his family members but his marketing by no means arrives.
Here is my web-site ... extended car auto warranty (Click at Frag5.com)