From Simple English Wikipedia, the free encyclopedia
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.
- See also triangle number. This is one of the most useful series: many applications can be found throughout mathematics.
![{\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7c3d53b94cffdf035061abe2c95c2bd73d3df4)
![{\displaystyle \sum _{i=1}^{n}i^{3}=\left[{\frac {n(n+1)}{2}}\right]^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}=\left(\sum _{i=1}^{n}i\right)^{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/559f22083e56d1f012dcf7bd7176a4af73435d0d)
![{\displaystyle \sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}={\frac {6n^{5}+15n^{4}+10n^{3}-n}{30}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11c3d371a1c417fd17fe7cd6c2e8fc24549ccea3)
- Where
is the
th Bernoulli number,
is negative and
is the binomial coefficient (choose function).
![{\displaystyle \sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}=\zeta (s)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75470e4fdc0e9e25bdc8da4577a3b2e072ce251f)
- Where
is the Riemann zeta function.
Infinite sum (for ) |
Finite sum
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where and
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where Lis(x) is the polylogarithm of x.
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![{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=\log _{e}\left({\frac {1}{1-x}}\right)\quad {\mbox{ for }}|x|<1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e005df60f7ea2b3528a18472110d78782bfe2aa0)
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots =\arctan(x)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db31613439934b8448183b52101ec959fba328b1)
![{\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}=\mathrm {arctanh} (x)\quad {\mbox{ for }}|x|<1\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd59bd1b9492608d782297ae1def4e1fbc246d2b)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f925fb6adb4df89b4b53d21814d20b744c168281)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {\pi ^{4}}{90}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aecd2aef6022fe0025c1c5d860ca95136283e607)
![{\displaystyle \sum _{n=1}^{\infty }{\frac {y}{n^{2}+y^{2}}}=-{\frac {1}{2y}}+{\frac {\pi }{2}}\coth(\pi y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5766c41cfa7ac2ef5f9a024f19a4da3b681d145a)
Many power series which arise from Taylor's theorem have a coefficient containing a factorial.
![{\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/011f8b2d7ff4c426b861d90d7a17bf4aaf2b7e90)
(c.f. mean of Poisson distribution)
(c.f. second moment of Poisson distribution)
![{\displaystyle \sum _{i=0}^{\infty }i^{3}{\frac {x^{i}}{i!}}=(x+3x^{2}+x^{3})e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ee6affc0492dba421f34bfa575284e60dc5d38)
![{\displaystyle \sum _{i=0}^{\infty }i^{4}{\frac {x^{i}}{i!}}=(x+7x^{2}+6x^{3}+x^{4})e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3fb0dbcbc13d14491763105a95715b2e04c073)
![{\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}x^{2i+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots =\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad339a6a625bc9201957d3c8f0edc10eb2c96443)
![{\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}x^{2i}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots =\cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/271cf11f7dab5d050a13fa700fce6ea2bd2528ef)
![{\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef4641aaaf4a728e5d9a05565a9f4a89b4c9c47)
![{\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf79ebd7871bb27972dc7f9c41bee47533c6fb5c)
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=\arcsin x\quad {\mbox{ for }}|x|<1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ed74ee23f499d612db04e42206fcd9f00bedde)
![{\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{4^{i}(i!)^{2}(2i+1)}}x^{2i+1}=\mathrm {arcsinh} (x)\quad {\mbox{ for }}|x|<1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45dad12852ca5fab4e8f76df6a2c123d7970c06e)
Geometric series:
![{\displaystyle (1+x)^{-1}={\begin{cases}\displaystyle \sum _{i=0}^{\infty }(-x)^{i}&|x|<1\\\displaystyle \sum _{i=1}^{\infty }-(x)^{-i}&|x|>1\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac777c9cd4f292ea100ff641a48f7cc1345b7e8e)
Binomial Theorem:
![{\displaystyle (a+x)^{n}={\begin{cases}\displaystyle \sum _{i=0}^{\infty }{\binom {n}{i}}a^{n-i}x^{i}&|x|\!<\!|a|\\\displaystyle \sum _{i=0}^{\infty }{\binom {n}{i}}a^{i}x^{n-i}&|x|\!>\!|a|\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a85ce496ef7153441eb72439b85289974be226e)
![{\displaystyle (1+x)^{\alpha }=\sum _{i=0}^{\infty }{\alpha \choose i}x^{i}\quad {\mbox{ for all }}|x|<1{\mbox{ and all complex }}\alpha \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6c1b4d83030831e97377b8e1fac9db3164d602)
- with generalized binomial coefficients
![{\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8b98f9bb608b66d81865df3817246606033a00)
Square root:
![{\displaystyle {\sqrt {1+x}}=\sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{(1-2i)i!^{2}4^{i}}}x^{i}\quad {\mbox{ for }}|x|<1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/043098662f2ac2328ee95ade7be436a4371ba88b)
Miscellaneous:
- [1]
![{\displaystyle \sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40370349acb98d4055a8b7b846c84edcd4367d3a)
- [1]
![{\displaystyle \sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}(1-{\sqrt {1-4x}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67acf74d182aae51978fe30a25aa401498253c1b)
- [1]
![{\displaystyle \sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac9408f0622ba8156286498f140d9593da158ea)
- [1]
![{\displaystyle \sum _{i=0}^{\infty }{2i+n \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}\left({\frac {1-{\sqrt {1-4x}}}{2x}}\right)^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4133e07f5a06ed6c9aa49721b4e67319d94c31c)
![{\displaystyle \sum _{i=0}^{n}{n \choose i}=2^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d44ff08e1b8abd60f2478029df07bed55f6be7)
![{\displaystyle \sum _{i=0}^{n}{n \choose i}a^{(n-i)}b^{i}=(a+b)^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab5b7cda6041463c6b49b7be79db7d3ba411e050)
![{\displaystyle \sum _{i=0}^{n}(-1)^{i}{n \choose i}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4aa6f60acab0874c64e34aa24a9c340d63c0ce4)
![{\displaystyle \sum _{i=0}^{n}{i \choose k}={n+1 \choose k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/092919a4f6bc16ffee6088db27a296d25392324c)
![{\displaystyle \sum _{i=0}^{n}{k+i \choose i}={k+n+1 \choose n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c3b4efc01a1d5af52c866332b8b759e063a695)
![{\displaystyle \sum _{i=0}^{r}{r \choose i}{s \choose n-i}={r+s \choose n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03cbc7b6a9d24b7311acb38d981e2ff55bec7c7a)
Sums of sines and cosines arise in Fourier series.
![{\displaystyle \sum _{i=1}^{n}\sin \left({\frac {i\pi }{n}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb4a07c73ffe63b6d69604d4359e15acff62100)
![{\displaystyle \sum _{i=1}^{n}\cos \left({\frac {i\pi }{n}}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b0a8e91ebcd4e42a6407b54598931099dcff66)
![{\displaystyle \sum _{n=b+1}^{\infty }{\frac {b}{n^{2}-b^{2}}}=\sum _{n=1}^{2b}{\frac {1}{2n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/718f9c96632a54d315b94c3af102b8317aa879c5)
- ↑ 1.0 1.1 1.2 1.3 Theoretical computer science cheat sheet