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| This '''list of mathematical series''' contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.
| | Τhere are far more people who wіsh tօ consume healthful as opposed to those who Ԁo. If you are looking to imprоve yօur ways of eating, then this articlе is perfеct for үou. Implementing far more wholesοme ways of eating is much easieг than you could possibly believe that. You don't have to transform yߋսr diet upside-down. The ideаs in the following paragrаphs will help you achieve outcomes easily.<br><br>Nutrition details labеling on well prepared foods bundles must be read meticulouslƴ. Even though one thing is brandеd "decreased-excess fat", it wіll be loaɗed with sugar, sodium, or somе other baɗ components. Ultra-processеd food needs to be avoided if you're attempting to losе weight. Labeling collection sսbstances within an item. If you aгe not really acquainted with the dіfferent technological vocabulary useful for various sugars and salts, seek advice from a dietіtian. You can also get a wealth оf ԁetails online in regards to what the language on food tags imply. Keep away from аny merchandise by using a laundry selection of man-made elements.<br><br>When yߋu arе expecting a baby or nursing you need to remain healthy. Women that are pregnant should invarіably be likely to try to eat sufficient health proteins. However, theƴ may not need tօ enjoу. Consider blending ovum in yօur early morning smootҺy for fսrther health proteins. Рregnant ѡomen should consider ovum as an exceрtional source of health proteins and а means to obtain а whߋlesߋme dish with lower energy with out fat. Always consume eggs that ɦavе been pasteurized to lesѕеn the potential risk of any medical problems.<br><br>An ordinarу product іs a functioning program. Ɗrink lоts օf ѡater and tгy and gеt adequate dietaгy fiber each day.<br><br>Specialists acknowledge that you have to minimize tɦe quantity of procesѕed flour and cereals in ƴour diet. Even so, eliminating these grains implies that yoս deprіving your self of fiber content and important nourishmеnt. Would it seem sensible to achieve thіs and aftеr that get whole wheat bacteria or fibeг content artificial additives tօ regenerate the advantages of natural wholegrain? Of course not.<br><br>Refined grains possess a mսch better taste, which is the reason packaged grain іs chаnging cereals. Utilizing bright white flour is much more efficient for some cooked mеrchandise. Even so, graіn typіcally ѕupply hеartier taste and also more dietary fiber to help you sսitable food Ԁigеstion.<br><br>To [http://www.Adobe.com/cfusion/search/index.cfm?term=&reduce+depression&loc=en_us&siteSection=home reduce depression] natuгally, eat food products loaded with Vitamin supplement B6. TҺis assists and also hardwearing . body's sеrotonin ranges on the ƿroper degreе, which could stop an discrepancy that frequently leads to major depression. Asparagus, poultrү and grain germ are all excellent sources of supplement B6. In the winter months, serotonin levels are evеn more ѕignificant.<br><br>Try having processed salmon. It is filled witҺ nutritional supplements your system needs աhich is low fat. Be sure to put selection to your diet to avoid dullness.<br><br>Pureеd berries, pears, or peaches make a quick and scrumptious ɡoody. Pureeing theѕe fгesh fruits produces a paste աhich makes a great topping or dip for a lot of diverse foods, frоm french fries to pretzels. Try different fruit alоng with distinct prep strategies [http://Anuncieaqui.eu/author/nudfr/ Vigrx Plus effects] to actually don't get tiгed of this snack fߋod.<br><br>Nutritional B12 іs excellent when уou aгe expectant because of its comforting advantages. Eating a good amount of food items lоaded with vitamin B12 can reԁuce tҺe hazards of creating some delivery flaws. B12 shortage isn't a standaгd difficulty. It's easier to be ѕafe tҺan sorry, even though.<br><br>Ensure each meal consume is balanced. Νutritional vitamins, body fat, and aminos all mɑke cօntributions to make a healthieг body. While you can get the vitamins and minerɑls in certain nutritional supplements, it's easier to buy them fгom meals.<br><br>A lߋt of people have issues identifying healthy food cҺoices. 7-grain a loɑf of bread will not have total-whole grains within it, therefore it iѕ less healthy because it appears to be. Be sure you consider the food label to ѕee what's truly in it. The packаging іs rarely a sіgn.<br><br>Hopefully you happen to be now in a poѕition to takе in greater by utilizing the easy-to-comply with methods that [http://Partyinradio.com/blog/groups/recommendations-on-excellent-nutrition-the-key-to-for-a-longer-time-existence/ vigrx Plus retailers] were gone οver here regarding your diet. Undеrstand that it's insufficient to niЬble on healthy only once in a while. The better consistent you will be, the greater number of constant your [http://escortlaradana.com/author/waveitch/ vigrx plus effects] will likely be. |
| *Here, <math>0^0</math> is taken to have the value 1.
| |
| *<math>B_n(x)</math> is a [[Bernoulli polynomial]].
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| *<math>B_n</math> is a [[Bernoulli number]], and here, <math>B_1=-\frac{1}{2}.</math>
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| *<math>E_n</math> is an [[Euler number]].
| |
| *<math>\zeta(s)</math> is the [[Riemann zeta function]].
| |
| *<math>\Gamma(z)</math> is the [[gamma function]].
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| *<math>\psi_n(z)</math> is a [[polygamma function]].
| |
| *<math>\operatorname{Li}_s(z)</math> is a [[polylogarithm]].
| |
| | |
| ==Sums of powers==
| |
| See [[Faulhaber's formula]].
| |
| *<math>\sum_{k=0}^m k^{n-1}=\frac{B_n(m+1)-B_n}{n}\,\!</math>
| |
| The first few values are:
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| *<math>\sum_{k=1}^m k=\frac{m(m+1)}{2}\,\!</math>
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| | |
| *<math>\sum_{k=1}^m k^2=\frac{m(m+1)(2m+1)}{6}=\frac{m^3}{3}+\frac{m^2}{2}+\frac{m}{6}\,\!</math>
| |
| | |
| *<math>\sum_{k=1}^m k^3 =\left[\frac{m(m+1)}{2}\right]^2=\frac{m^4}{4}+\frac{m^3}{2}+\frac{m^2}{4}\,\!</math>
| |
| | |
| See [[zeta constants]].
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| *<math>\zeta(2n)=\sum^{\infty}_{k=1} \frac{1}{k^{2n}}=(-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!} </math>
| |
| The first few values are:
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| *<math>\zeta(2)=\sum^{\infty}_{k=1} \frac{1}{k^2}=\frac{\pi^2}{6}\,\!</math> (the [[Basel problem]])
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| *<math>\zeta(4)=\sum^{\infty}_{k=1} \frac{1}{k^4}=\frac{\pi^4}{90}\,\!</math>
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| *<math>\zeta(6)=\sum^{\infty}_{k=1} \frac{1}{k^6}=\frac{\pi^6}{945}\,\!</math>
| |
| | |
| ==Power series==
| |
| | |
| ===Low-order polylogarithms===
| |
| Finite sums:
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| *<math>\sum_{k=0}^{n} z^k = \frac{1-z^{n+1}}{1-z}\,\!</math>, ([[geometric series]])
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| | |
| *<math>\sum_{k=1}^n k z^k = z\frac{1-(n+1)z^n+nz^{n+1}}{(1-z)^2}\,\!</math>
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| *<math>\sum_{k=1}^n k^2 z^k = z\frac{1+z-(n+1)^2z^n+(2n^2+2n-1)z^{n+1}-n^2z^{n+2}}{(1-z)^3} \,\!</math>
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| *<math>\sum_{k=1}^n k^m z^k = \left(z \frac{d}{dz}\right)^m \frac{z-z^{n+1}}{1-z}</math>
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| | |
| Infinite sums, valid for <math>|z|<1</math> (see [[polylogarithm]]):
| |
| *<math>\operatorname{Li}_n(z)=\sum_{k=1}^{\infty} \frac{z^k}{k^n}\,\!</math>
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| The following is a useful property to calculate low-integer-order polylogarithms recursively in [[Closed-form expression|closed form]]:
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| *<math>\frac{d}{dz}\operatorname{Li}_n(z)=\frac{\operatorname{Li}_{n-1}(z)}{z}\,\!</math>
| |
| | |
| *<math>\operatorname{Li}_{1}(z)=\sum_{k=1}^\infty \frac{z^k}{k}=-\ln(1-z)\!</math>
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| | |
| *<math>\operatorname{Li}_{0}(z)=\sum_{k=1}^\infty z^k=\frac{z}{1-z}\!</math>
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| | |
| *<math>\operatorname{Li}_{-1}(z)=\sum_{k=1}^\infty k z^k=\frac{z}{(1-z)^2}\,\!</math>
| |
| | |
| *<math>\operatorname{Li}_{-2}(z)=\sum_{k=1}^\infty k^2 z^k=\frac{z(1+z)}{(1-z)^3}\,\!</math>
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| | |
| *<math>\operatorname{Li}_{-3}(z)=\sum_{k=1}^\infty k^3 z^k =\frac{z(1+4z+z^2)}{(1-z)^4}\,\!</math>
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| | |
| *<math>\operatorname{Li}_{-4}(z)=\sum_{k=1}^\infty k^4 z^k =\frac{z(1+z)(1+10z+z^2)}{(1-z)^5}\,\!</math>
| |
| | |
| ===Exponential function===
| |
| *<math>\sum_{k=0}^\infty \frac{z^k}{k!} = e^z\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty k\frac{z^k}{k!} = z e^z\,\!</math> (cf. mean of [[Poisson distribution]])
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| | |
| *<math>\sum_{k=0}^\infty k^2 \frac{z^k}{k!} = (z + z^2) e^z\,\!</math> (cf. [[second moment]] of Poisson distribution)
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| | |
| *<math>\sum_{k=0}^\infty k^3 \frac{z^k}{k!} = (z + 3z^2 + z^3) e^z\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty k^4 \frac{z^k}{k!} = (z + 7z^2 + 6z^3 + z^4) e^z\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty k^n \frac{z^k}{k!} = z \frac{d}{dz} \sum_{k=0}^\infty k^{n-1} \frac{z^k}{k!}\,\! = e^z T_{n}(z) </math>
| |
| | |
| where <math>T_{n}(z)</math> is the [[Touchard polynomials]].
| |
| | |
| ===Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions===
| |
| *<math>\sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}=\sin z\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)!}=\sinh z\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!}=\cos z\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{z^{2k}}{(2k)!}=\cosh z\,\!</math>
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| *<math>\sum_{k=1}^\infty \frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tan z, |z|<\frac{\pi}{2}\,\!</math>
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| *<math>\sum_{k=1}^\infty \frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tanh z, |z|<\frac{\pi}{2}\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{(-1)^k2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\cot z, |z|<\pi\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\coth z, |z|<\pi\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\csc z, |z|<\pi\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\operatorname{csch} z, |z|<\pi\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{(-1)^kE_{2k}z^{2k}}{(2k)!}=\sec z, |z|<\frac{\pi}{2}\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{E_{2k}z^{2k}}{(2k)!}=\operatorname{sech} z, |z|<\frac{\pi}{2}\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\arcsin z, |z|\le1\,\!</math>
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| | |
| *<math>\sum_{k=0}^\infty \frac{(-1)^k(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\operatorname{arsinh} {z}, |z|\le1\,\!</math>
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| *<math>\sum_{k=0}^\infty \frac{(-1)^kz^{2k+1}}{2k+1}=\arctan z, |z|<1\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{z^{2k+1}}{2k+1}=\operatorname{arctanh} z, |z|<1\,\!</math>
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| *<math>\ln2+\sum_{k=1}^\infty \frac{(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^2}=\ln\left(1+\sqrt{1+z^2}\right), |z|\le1\,\!</math>
| |
| | |
| ===Modified-factorial denominators===
| |
| *<math>\sum^{\infty}_{k=0} \frac{(4k)!}{2^{4k} \sqrt{2} (2k)! (2k+1)!} z^k = \sqrt{\frac{1-\sqrt{1-z}}{z}}, |z|<1</math><ref name="gfo">[http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf generatingfunctionology]</ref>
| |
| | |
| *<math>\sum^{\infty}_{k=0} \frac{2^{2k} (k!)^2}{(k+1) (2k+1)!} z^{2k+2} = \left(\arcsin{z}\right)^2, |z|\le1 </math><ref name="gfo"/>
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| | |
| *<math>\sum^{\infty}_{n=0} \frac{\prod_{k=0}^{n-1}(4k^2+\alpha^2)}{(2n)!} z^{2n} + \sum^{\infty}_{n=0} \frac{\alpha \prod_{k=0}^{n-1}[(2k+1)^2+\alpha^2]}{(2n+1)!} z^{2n+1} = e^{\alpha \arcsin{z}}, |z|\le1 </math>
| |
| | |
| === Binomial coefficients ===
| |
| *<math>(1+z)^\alpha = \sum_{k=0}^\infty {\alpha \choose k} z^k, |z|<1</math> (see [[Binomial theorem]])
| |
| *<ref name="ctcs">[http://www.tug.org/texshowcase/cheat.pdf Theoretical computer science cheat sheet]</ref> <math>\sum_{k=0}^\infty {{\alpha+k-1} \choose k} z^k = \frac{1}{(1-z)^\alpha}, |z|<1</math>
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| *<ref name="ctcs"/> <math>\sum_{k=0}^\infty \frac{1}{k+1}{2k \choose k} z^k = \frac{1-\sqrt{1-4z}}{2z}, |z|<\frac{1}{4}</math>, generating function of the [[Catalan numbers]]
| |
| | |
| *<ref name="ctcs"/> <math>\sum_{k=0}^\infty {2k \choose k} z^k = \frac{1}{\sqrt{1-4z}}, |z|<\frac{1}{4}</math>, generating function of the [[Central binomial coefficient]]s
| |
| | |
| *<ref name="ctcs"/> <math>\sum_{k=0}^\infty {2k + \alpha \choose k} z^k = \frac{1}{\sqrt{1-4z}}\left(\frac{1-\sqrt{1-4z}}{2z}\right)^\alpha, |z|<\frac{1}{4}</math>
| |
| | |
| ===[[Harmonic number]]s===
| |
| *<math> \sum_{k=1}^\infty H_k z^k = \frac{-\ln(1-z)}{1-z}, |z|<1</math>
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| *<math> \sum_{k=1}^\infty \frac{H_k}{k+1} z^{k+1} = \frac{1}{2}\left[\ln(1-z)\right]^2, \qquad |z|<1</math>
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| *<math> \sum_{k=1}^\infty \frac{(-1)^{k-1} H_{2k}}{2k+1} z^{2k+1} = \frac{1}{2} \arctan{z} \log{(1+z^2)}, \qquad |z|<1 </math><ref name="gfo"/>
| |
| | |
| *<math> \sum_{n=0}^\infty \sum_{k=0}^{2n} \frac{(-1)^k}{2k+1} \frac{z^{4n+2}}{4n+2} = \frac{1}{4} \arctan{z} \log{\frac{1+z}{1-z}},\qquad |z|<1 </math><ref name="gfo"/>
| |
| | |
| == Binomial coefficients ==
| |
| *<math>\sum_{k=0}^n {n \choose k} = 2^n</math>
| |
| | |
| *<math>\sum_{k=0}^n (-1)^k {n \choose k} = 0</math>
| |
| | |
| *<math>\sum_{k=0}^n {k \choose m} = { n+1 \choose m+1 }</math>
| |
| | |
| *<math>\sum_{k=0}^n {m+k-1 \choose k} = { n+m \choose n }</math> (see [[Multiset#Recurrence relation|Multiset]])
| |
| | |
| *<math>\sum_{k=0}^n {\alpha \choose k}{\beta \choose n-k} = {\alpha+\beta \choose n}</math> (see [[Vandermonde identity]])
| |
| | |
| == Trigonometric functions ==
| |
| Sums of [[sine]]s and [[cosine]]s arise in [[Fourier series]].
| |
| | |
| *<math>\sum_{k=1}^\infty \frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2}, 0<\theta<2\pi\,\!</math>
| |
| | |
| *<math>\sum_{k=1}^\infty \frac{\cos(k\theta)}{k}=-\frac{1}{2}\ln(2-2\cos\theta), \theta\in\mathbb{R}\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^\infty \frac{\sin[(2k+1)\theta]}{2k+1}=\frac{\pi}{4}, 0<\theta<\pi\,\!</math>
| |
| | |
| *<math>B_n(x)=-\frac{n!}{2^{n-1}\pi^n}\sum_{k=1}^\infty \frac{1}{k^n}\cos\left(2\pi kx-\frac{\pi n}{2}\right), 0<x<1\,\!</math><ref>{{cite web|url=http://functions.wolfram.com/Polynomials/BernoulliB2/06/02/|title=Bernoulli polynomials: Series representations (subsection 06/02)|accessdate=2 June 2011}}</ref>
| |
| | |
| *<math>\sum_{k=0}^n \sin(\theta+k\alpha)=\frac{\sin\frac{(n+1)\alpha}{2}\sin(\theta+\frac{n\alpha}{2})}{\sin\frac{\alpha}{2}}\,\!</math>
| |
| | |
| *<math>\sum_{k=1}^{n-1} \sin\frac{\pi k}{n}=\cot\frac{\pi}{2n}\,\!</math>
| |
| | |
| *<math>\sum_{k=1}^{n-1} \sin\frac{2\pi k}{n}=0\,\!</math>
| |
| | |
| *<math>\sum_{k=0}^{n-1} \csc^2\left(\theta+\frac{\pi k}{n}\right)=n^2\csc^2(n\theta)\,\!</math><ref>{{cite web|last=Hofbauer|first=Josef|title=A simple proof of 1+1/2^2+1/3^2+...=PI^2/6 and related identities|url=http://homepage.univie.ac.at/josef.hofbauer/02amm.pdf|accessdate=2 June 2011}}</ref>
| |
| | |
| *<math>\sum_{k=1}^{n-1} \csc^2\frac{\pi k}{n}=\frac{n^2-1}{3}\,\!</math>
| |
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| *<math>\sum_{k=1}^{n-1} \csc^4\frac{\pi k}{n}=\frac{n^4+10n^2-11}{45}\,\!</math>
| |
| | |
| == Rational functions ==
| |
| *<math>\sum_{m=b+1}^{\infty} \frac{b}{m^2 - b^2} = \frac{1}{2} H_{2b}</math>
| |
| | |
| *<math>\sum^{\infty}_{m=1} \frac{y}{m^2+y^2} = -\frac{1}{2y}+\frac{\pi}{2}\coth(\pi y)</math><ref>[[Eric W. Weisstein|Weisstein, Eric W.]], "[http://mathworld.wolfram.com/RiemannZetaFunction.html Riemann Zeta Function]" from [[MathWorld]], equation 52</ref>
| |
| | |
| *An infinite series of any [[rational function]] of <math>n</math> can be reduced to a finite series of [[polygamma function]]s, by use of [[partial fraction decomposition]].<ref>[http://people.math.sfu.ca/~cbm/aands/ Abramowitz and Stegun]</ref> This fact can also be applied to finite series of rational functions, allowing the result to be computed in [[constant time]] even when the series contains a large number of terms.
| |
| | |
| ==See also==
| |
| {{Div col|cols=3}}
| |
| * [[Series (mathematics)]]
| |
| * [[List of integrals]]
| |
| * [[Summation#Identites|Summation]]
| |
| * [[Taylor series]]
| |
| * [[Binomial theorem]]
| |
| * [[Gregory's series]]
| |
| * [[On-Line Encyclopedia of Integer Sequences]]
| |
| {{Div col end}}
| |
| | |
| ==Notes==
| |
| {{Reflist|30em}}
| |
| | |
| ==References==
| |
| *Many books with a [[list of integrals]] also have a list of series.
| |
| | |
| [[Category:Mathematical series]]
| |
| [[Category:Mathematics-related lists|Series]]
| |
| [[Category:Mathematical tables|Series]]
| |
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