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<br><br>Christmas is soon to come and everybody loves celebrating this season. It is the time of 2010 where everyone can expresses their love and kind thoughts by giving and exchanging gifts to and another. The falling of the snowflakes, Yuletide celebration, Christmas shopping, using a talk with guests in a relax and warm fireplace with hot cups of chocolate drink added with the deliciously prepared food basically are the things to look forward to in this particular season. For people who always love fashion and trends, it is usually one exciting member of the season to be updated in achievable will give you style and fashion to come in the season. The following paragraphs will give tips and knowledge about the latest fashion and style in this winter weather.<br><br>Dangle Earrings are always the craze this season. They attract instant awareness to your face and neck, no matter their size, shape or color. Use them with indictment.<br><br>Valentines Day is the suitable opportunity for ladies to show their gratitude for their significant some other. Men are always there to save the day when a car tire needs changing, any stubborn jar won't open, when we want a shoulder to cry on following a bad holiday to work, in order to make late night runs to your grocery store when you've run out of milk. They put up just about all the sorts of nagging year round so with romantic holiday of the majority coming up, it 's time to show some thanks. Accept is as true or not, women are not the only ones who adore receiving jewellery . There are plenty of choices for giving jewellery to men on Love day. Below are some of the appealing Love gifts just give or perhaps a man can ever have.<br><br>The same goes for bathrooms. Excluding the ambient lighting, the mirrors in bathroom are usually clear and free of glare, when soft-glowing wall sconces were installed on either side of that company. Bath tubs can brighten up when a correctly enclosed Tiffany Lamp is placed beside these kind of. And also, don't restrict lighting to for recycling paper ways. The of lights or the placement of lights can be explored.<br><br>Don't chuck your girlfriend's age and personality. A younger woman in her 20s is lively and bubbly; thus, a pink or sky blue pendant is prime. For older women on one other hand, black Pendants or red, will obviously make them feel appreciated and esteemed. Jewelry is crucial in every girl's lifestyle because they compliment a dressing up. Regardless of your price, accessories need to function well along with a skirt, a blouse perhaps with eliminating of your sweetheart's look.<br><br>June - Creamy Pearl Moonstones: This stone represents someone with generosity. Are usually a peaceful person and seek the gentle things in life span. Your ability to the touch people's lives changes who they really are for far better.<br><br>I am always adding new jewelry creations to my Etsy site. In addition, I'm a member of crafthaus, a broadband social network/community of contemporary artists most of my work, including vessels and conceptual pieces, can be located on that website.<br><br>If you have any thoughts pertaining to in which and how to use [https://www.youtube.com/watch?v=pns2W-mSPU8 Quantum Pendants], you can contact us at our site.
{{Calculus |Series}}
 
In [[mathematics]], an '''alternating series''' is an [[infinite series]] of the form
 
:<math>\sum_{n=0}^\infty (-1)^n\,a_n</math> or <math>\sum_{n=1}^\infty (-1)^{n-1}\,a_n</math>
 
with ''a<sub>n</sub>'' > 0 for all&nbsp;''n''. The signs of the general terms alternate between positive and negative. Like any series, an alternating [[Convergent series|series converges]] if and only if the associated sequence of partial sums [[Limit of a sequence|converges]].
 
== Alternating series test ==
{{main|alternating series test}}
 
The theorem known as "Leibniz Test" or the [[alternating series test]] tells us that an alternating series will converge if the terms ''a<sub>n</sub>'' converge to 0 [[monotonic function|monotonically]].
 
Proof: Suppose the sequence <math>a_n</math> converges to zero and is monotone decreasing. If <math>m</math> is odd and <math>m<n</math>, we obtain the estimate <math>S_m - S_n < a_{m}</math> via the following calculation:
 
: <math>
\begin{align}
S_m - S_n & =
\sum_{k=0}^m(-1)^k\,a_k\,-\,\sum_{k=0}^n\,(-1)^k\,a_k\ = \sum_{k=m+1}^n\,(-1)^k\,a_k  \\
& =a_{m+1}-a_{m+2}+a_{m+3}-a_{m+4}+\cdots+a_n\\
& =\displaystyle a_{m+1}-(a_{m+2}-a_{m+3}) - (a_{m+4}-a_{m+5}) -\cdots-a_n \le a_{m+1}\le a_{m}  [a_{n} decreasing].
\end{align}
</math>
 
Since <math>a_n</math> is monotonically decreasing, the terms <math>-(a_m - a_{m+1})</math> are negative. Thus, we have the final inequality <math>S_m - S_n \le a_{m}</math> .Similarly it can be shown that <math>-a_{m}\le S_m - S_n </math>.   Since <math>a_{m}</math> converges to <math>0</math>, our partial sums <math>S_m</math> form a [[Cauchy sequence]] (i.e. the series satisfies the [[Cauchy convergence criterion for series]]) and therefore converge. The argument for <math>m</math> even is similar.
 
== Approximating sums ==
The estimate above does not depend on <math>n</math>. So, if <math>a_n</math> is approaching 0 monotonically, the estimate provides an [[error bound]] for approximating infinite sums by partial sums:
 
: <math>|\sum_{k=0}^\infty(-1)^k\,a_k\,-\,\sum_{k=0}^m\,(-1)^k\,a_k|\le |a_{m+1}|.</math>
 
== Absolute convergence ==
A series <math>\sum a_n</math> [[absolute convergence|converges absolutely]] if the series <math>\sum |a_n|</math> converges.
 
Theorem: Absolutely convergent series are convergent.
 
Proof: Suppose <math>\sum a_n</math> is absolutely convergent. Then, <math>\sum |a_n|</math> is convergent and it follows that <math>\sum 2|a_n|</math> converges as well. Since <math> 0 \leq a_n + |a_n| \leq 2|a_n|</math>, the series <math>\sum (a_n + |a_n|)</math> converges by the [[comparison test]]. Therefore, the series <math>\sum a_n</math> converges as the difference of two convergent series <math>\sum a_n = \sum (a_n + |a_n|) - \sum |a_n|</math>.
 
== Conditional convergence ==
A series is [[Conditional convergence|conditionally convergent]] if it converges but does not converge absolutely.
 
For example, the [[harmonic series (mathematics)|harmonic series]]
 
:<math>\sum_{n=1}^\infty \frac{1}{n},\! </math>
 
diverges, while the alternating version
 
:<math>\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n},\! </math>
 
converges by the [[Alternating_series#Alternating_series_test|alternating series test]].
 
== Rearrangements ==
For any series, we can create a new series by rearranging the order of summation. A series is [[Series_(mathematics)#Unconditionally_convergent_series|unconditionally convergent]] if any rearrangement creates a series with the same convergence as the original series. [[Absolute_convergence#Rearrangements_and_unconditional_convergence|Absolutely convergent series are unconditionally convergent]]. But the [[Riemann series theorem]] states that conditionally convergent series can be rearranged to create arbitrary convergence.<ref>{{cite journal |last1=Mallik |first1=AK|year=2007 |title=Curious Consequences of Simple Sequences |journal=Resonance |volume=12 |issue=1 |pages=23–37 |url=http://www.springerlink.com/index/D65WX2N5384880LV.pdf}}</ref> The general principle is that addition of infinite sums is only commutative for absolutely convergent series.
 
For example, this [[Mathematical_fallacy#Associative_law|false proof that 1=0]] exploits the failure of associativity for infinite sums.
 
As another example, [[Natural_logarithm#Derivative.2C_Taylor_series|we know that]]
:<math>\ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.</math>
 
But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for <math>\frac{1}{2}\ln(2)</math>:
 
:<math>
\begin{align}
& {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4}+\left(\frac{1}{3}-\frac{1}{6}\right)-\frac{1}{8}+\left(\frac{1}{5}-\frac{1}{10}\right)-\frac{1}{12}+\cdots \\[8pt]
& = \frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}+\cdots \\[8pt]
& = \frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots\right)= \frac{1}{2} \ln(2).
\end{align}
</math>
 
Another valid example of alternating series is the following
:<math>
\sum_{k=0}^{\infty}
\frac{(-1)^{k}}{\sqrt{k+1}}=1-\frac{1}{\sqrt{2}}
+\frac{1}{\sqrt{3}}
-\frac{1}{\sqrt{4}}
+\frac{1}{\sqrt{5}}
\cdots=-(\sqrt{2}
-1)\zeta(\frac{1}{2})\approx0.6048986434....
</math>
 
== Series acceleration ==
In practice, the numerical summation of an alternating series may be sped up using any one of a variety of [[series acceleration]] techniques. One of the oldest techniques is that of [[Euler summation]], and there are many modern techniques that can offer even more rapid convergence.
 
==See also==
* [[Nörlund–Rice integral]]
* [[Series (mathematics)]]
 
==Notes==
{{reflist}}
 
==References==
*{{MathWorld|title=Alternating Series|urlname=AlternatingSeries}}
 
{{DEFAULTSORT:Alternating Series}}
[[Category:Calculus]]
[[Category:Mathematical series]]
[[Category:Real analysis]]

Revision as of 12:59, 15 February 2014



Christmas is soon to come and everybody loves celebrating this season. It is the time of 2010 where everyone can expresses their love and kind thoughts by giving and exchanging gifts to and another. The falling of the snowflakes, Yuletide celebration, Christmas shopping, using a talk with guests in a relax and warm fireplace with hot cups of chocolate drink added with the deliciously prepared food basically are the things to look forward to in this particular season. For people who always love fashion and trends, it is usually one exciting member of the season to be updated in achievable will give you style and fashion to come in the season. The following paragraphs will give tips and knowledge about the latest fashion and style in this winter weather.

Dangle Earrings are always the craze this season. They attract instant awareness to your face and neck, no matter their size, shape or color. Use them with indictment.

Valentines Day is the suitable opportunity for ladies to show their gratitude for their significant some other. Men are always there to save the day when a car tire needs changing, any stubborn jar won't open, when we want a shoulder to cry on following a bad holiday to work, in order to make late night runs to your grocery store when you've run out of milk. They put up just about all the sorts of nagging year round so with romantic holiday of the majority coming up, it 's time to show some thanks. Accept is as true or not, women are not the only ones who adore receiving jewellery . There are plenty of choices for giving jewellery to men on Love day. Below are some of the appealing Love gifts just give or perhaps a man can ever have.

The same goes for bathrooms. Excluding the ambient lighting, the mirrors in bathroom are usually clear and free of glare, when soft-glowing wall sconces were installed on either side of that company. Bath tubs can brighten up when a correctly enclosed Tiffany Lamp is placed beside these kind of. And also, don't restrict lighting to for recycling paper ways. The of lights or the placement of lights can be explored.

Don't chuck your girlfriend's age and personality. A younger woman in her 20s is lively and bubbly; thus, a pink or sky blue pendant is prime. For older women on one other hand, black Pendants or red, will obviously make them feel appreciated and esteemed. Jewelry is crucial in every girl's lifestyle because they compliment a dressing up. Regardless of your price, accessories need to function well along with a skirt, a blouse perhaps with eliminating of your sweetheart's look.

June - Creamy Pearl Moonstones: This stone represents someone with generosity. Are usually a peaceful person and seek the gentle things in life span. Your ability to the touch people's lives changes who they really are for far better.

I am always adding new jewelry creations to my Etsy site. In addition, I'm a member of crafthaus, a broadband social network/community of contemporary artists most of my work, including vessels and conceptual pieces, can be located on that website.

If you have any thoughts pertaining to in which and how to use Quantum Pendants, you can contact us at our site.

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