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{{Calculus |Series}}
== as if a huge tip ...... now instantly spin up into a river ==


The '''alternating series test''' is a method used to prove that infinite [[series (mathematics)|series]] of terms  [[Convergent series|converge]]. It was discovered by [[Gottfried Leibniz]] and is sometimes known as '''Leibniz's test''' or the '''Leibniz criterion'''.
Circles beast Mohe look forward to,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_29.htm オークリー サングラス ノーズパッド].<br><br>root of this war, but sector Mohe and Mo Luosa beast who was finally able to live!<br><br>'Star Tower.'<br><br>Luo Feng suddenly roar and saw appeared out of thin air in front of a nine-story tower,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_68.htm ゴルフ オークリー サングラス], the tower is the stars,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_77.htm オークリー サングラス 通販], I saw stars tower sideways,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_18.htm オークリー サングラス フラックジャケット], as if a huge tip ...... now instantly spin up into a river,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリー サングラス ジャパンフィット], which is a golden directly toward the two winding river channel swirl impact in the past.<br><br>this trick is Luo Feng Twelve bands own top secret method, the study and understanding of the stars of the latest heavy Arcane Tower,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー サングラス ジョウボーン], Luo Feng was named 'Citizen of the river.'<br><br>'River of Stars' on 'cycle of destruction'!<br><br>'Peng!'<br>golden river of dazzling moments<br>collapsing, the stars were bombarded throughout the tower tumbling throw fly far away,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリー サングラス ジャパンフィット], and two whirlpool channel power and influence diminished almost straight Luo Feng this,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_21.htm 登山 サングラス オークリー], but if look carefully, these two swirl channels the outer wall of the channel already has a lot of cracks.
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<ul>
 
  <li>[http://www.dtytwh.com/home.php?mod=space&uid=289124 http://www.dtytwh.com/home.php?mod=space&uid=289124]</li>
 
  <li>[http://bbs.guifucaifu.com/home.php?mod=space&uid=236435 http://bbs.guifucaifu.com/home.php?mod=space&uid=236435]</li>
 
  <li>[http://www2u.biglobe.ne.jp/~c_steam/turedure/petit.cgi http://www2u.biglobe.ne.jp/~c_steam/turedure/petit.cgi]</li>
 
</ul>


== Formulation ==
== sitting on the sofa. ==


A series of the form
'This book a total of 139 points,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_57.htm オークリー サングラス レンズ交換], I have to draw you a question about.' Luo Feng, handing the history books in the hands of his brother Rituximab heard, could not help a smile took over: 'Go ahead, have a rare opportunity to quiz Your brother, brother you listen, I'm going out of the question you answer wrong, you can lose face. '<br><br>'Go ahead.' Luo Feng smiled,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_7.htm オークリー サングラス], sitting on the sofa.<br><br>'once one man,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_51.htm 激安オークリーサングラス], in the Yangtze River,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm オークリー サングラスケース], beheaded 'tiger strange Kau', save hundreds of thousands of people,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_23.htm オークリーサングラス ランキング], hundreds of thousands of people make a successful move to the southern base of heroic sacrifice is who? Where? age at death? detailed date and time did it happen? 'Rituximab look a bit closer, then a question.<br><br>'heroic sacrifice,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_73.htm オークリー サングラス 人気], was awarded four stars by the national hero medal chase 'Dong Nanbiao', he is the former Jiangsu Taixing people. died aged 39,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_73.htm オークリー サングラス 店舗], detailed date and time occurs,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_63.htm ゴルフ サングラス オークリー], it should be the year 2018 ......' Luo Feng could not help wrinkle from his brow.<br><br>brother Rituximab, help chase
 
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:<math> \sum_{n=1}^\infty (-1)^{n-1} a_n = a_1 - a_2 + a_3 - \cdots \!</math>
<ul>
 
 
Or,
  <li>[http://wurihome.com/home.php?mod=space&uid=17502 http://wurihome.com/home.php?mod=space&uid=17502]</li>
 
 
:<math> \sum_{n=1}^\infty (-1)^{n} a_n = - a_1 + a_2 - a_3 + \cdots \!</math>
  <li>[http://www.rieo.net/bbs/rieobbs.cgi http://www.rieo.net/bbs/rieobbs.cgi]</li>
 
 
where ''a''<sub>''n''</sub> are positive, is called an [[alternating series]].  
  <li>[http://qiyeneixun.org/plus/feedback.php?aid=1671 http://qiyeneixun.org/plus/feedback.php?aid=1671]</li>
 
 
The '''alternating series test''' then says if {''a''<sub>''n''</sub>} decreases monotonically and goes to 0 in the limit then the alternating series converges.
</ul>
 
Moreover, let ''L'' denote the sum of the series, then the partial sum
 
:<math>S_k = \sum_{n=1}^k (-1)^{n-1} a_n\!</math>
 
approximates ''L'' with error bounded by the next omitted term:
 
:<math>\left | S_k - L \right \vert \le \left | S_k - S_{k+1} \right \vert = a_{k+1}.\!</math>
 
==Proof <ref> The proof follows the idea given by James Stewart (2012) “Calculus: Early Transcendentals, Seventh Edition” pp. 727–730. ISBN 0-538-49790-4</ref>==
 
Suppose we are given a series of the form <math>\sum_{n=1}^\infty (-1)^{n-1} a_n\!</math>, where <math> \lim_{n\rightarrow\infty}a_{n}=0 </math> and <math> a_n \geq a_{n+1} </math> for all natural numbers ''n''. (The case <math>\sum_{n=1}^\infty (-1)^{n} a_n\!</math> follows by taking the negative.)
 
===Proof of convergence===
We will prove that both the partial sums <math>S_{2m+1}=\sum_{n=1}^{2m+1} (-1)^{n-1} a_n</math> with odd number of terms, and  <math>S_{2m}=\sum_{n=1}^{2m} (-1)^{n-1} a_n</math> with even number of terms, converge to the same number ''L''. Thus the usual partial sum <math>S_k=\sum_{n=1}^k (-1)^{n-1} a_n</math> also converges to ''L''.
 
The odd partial sums decrease monotonically:
 
:<math> S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3} \leq S_{2m+1} </math>
 
while the even partial sums increase monotonically:
 
:<math> S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2} \geq S_{2m} </math>
 
both because ''a''<sub>''n''</sub> decrease monotonically with ''n''.  
 
Moreover, since ''a''<sub>''n''</sub> are positive, <math> S_{2m+1}-S_{2m}=a_{2m+1} \geq 0 </math>. Thus we can collect these facts to form the following suggestive inequality:
 
:<math> a_1 - a_2 = S_2 \leq S_{2m} < S_{2m+1} \leq S_1 = a_1. </math>
 
Now, note that ''a''<sub>''1''</sub> - ''a''<sub>''2''</sub> is a lower bound of the monotonically decreasing sequence ''S''<sub>''2m+1''</sub>, [[monotone convergence theorem]] then implies that this sequence converges as ''m'' approaches infinity. Similarly, the sequence of even partial sum converges too.
 
Finally, they must converge to the same number because
 
:<math> \lim_{n\to\infty}S_{2m+1}-S_{2m}=\lim_{n\to\infty}a_{2m+1}=0. </math>
 
Call the limit ''L'', then the [[monotone convergence theorem]] also tells us an extra information that
 
:<math> S_{2m} \leq L \leq S_{2m+1} </math>
 
for any ''m''. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there are odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.  
 
This understanding leads immediately to an error bound of partial sums, shown below.
 
===Proof of partial sum error bound===
 
We would like to show <math>\left| S_k - L \right| \leq a_{k+1}\!</math> by splitting into two cases.
 
When k = 2m+1, i.e. odd, then
 
:<math>\left| S_{2m+1} - L \right| = S_{2m+1} - L \leq S_{2m+1} - S_{2m+2} = a_{(2m+1)+1} </math>
 
When k = 2m, i.e. even, then
 
:<math>\left| S_{2m} - L \right| = L - S_{2m} \leq S_{2m+1} - S_{2m} = a_{2m+1} </math>
 
as desired.  
 
Both cases rely essentially on the last inequality derived in the previous proof.
 
For an alternative proof using [[Cauchy's convergence test]], see [[Alternating series]].
 
For a generalization, see [[Dirichlet's test]].
 
== See also ==
*[[Alternating series]]
*[[Dirichlet's test]]
 
== Literature ==
 
* Knopp, Konrad, ''Infinite Sequences and Series'', Dover publications, Inc.,  New York, 1956. (§ 3.4) ISBN 0-486-60153-6
 
* Whittaker, E. T., and Watson, G. N., ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1963. (§ 2.3) ISBN 0-521-58807-3
 
==References==
 
<references/>
 
==External links==
* {{MathWorld | title=Leibniz Criterion | urlname=LeibnizCriterion}}
 
{{Gottfried Wilhelm Leibniz}}
 
{{DEFAULTSORT:Alternating Series Test}}
[[Category:Mathematical series]]
[[Category:Convergence tests]]
[[Category:Gottfried Leibniz]]

Latest revision as of 13:49, 12 October 2014

as if a huge tip ...... now instantly spin up into a river

Circles beast Mohe look forward to,オークリー サングラス ノーズパッド.

root of this war, but sector Mohe and Mo Luosa beast who was finally able to live!

'Star Tower.'

Luo Feng suddenly roar and saw appeared out of thin air in front of a nine-story tower,ゴルフ オークリー サングラス, the tower is the stars,オークリー サングラス 通販, I saw stars tower sideways,オークリー サングラス フラックジャケット, as if a huge tip ...... now instantly spin up into a river,オークリー サングラス ジャパンフィット, which is a golden directly toward the two winding river channel swirl impact in the past.

this trick is Luo Feng Twelve bands own top secret method, the study and understanding of the stars of the latest heavy Arcane Tower,オークリー サングラス ジョウボーン, Luo Feng was named 'Citizen of the river.'

'River of Stars' on 'cycle of destruction'!

'Peng!'
golden river of dazzling moments
collapsing, the stars were bombarded throughout the tower tumbling throw fly far away,オークリー サングラス ジャパンフィット, and two whirlpool channel power and influence diminished almost straight Luo Feng this,登山 サングラス オークリー, but if look carefully, these two swirl channels the outer wall of the channel already has a lot of cracks. 相关的主题文章：

sitting on the sofa.

'This book a total of 139 points,オークリー サングラス レンズ交換, I have to draw you a question about.' Luo Feng, handing the history books in the hands of his brother Rituximab heard, could not help a smile took over: 'Go ahead, have a rare opportunity to quiz Your brother, brother you listen, I'm going out of the question you answer wrong, you can lose face. '

'Go ahead.' Luo Feng smiled,オークリー サングラス, sitting on the sofa.

'once one man,激安オークリーサングラス, in the Yangtze River,オークリー サングラスケース, beheaded 'tiger strange Kau', save hundreds of thousands of people,オークリーサングラス ランキング, hundreds of thousands of people make a successful move to the southern base of heroic sacrifice is who? Where? age at death? detailed date and time did it happen? 'Rituximab look a bit closer, then a question.

'heroic sacrifice,オークリー サングラス 人気, was awarded four stars by the national hero medal chase 'Dong Nanbiao', he is the former Jiangsu Taixing people. died aged 39,オークリー サングラス 店舗, detailed date and time occurs,ゴルフ サングラス オークリー, it should be the year 2018 ......' Luo Feng could not help wrinkle from his brow.

brother Rituximab, help chase 相关的主题文章：

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