Aşağıdaki matematiksel seriler listesi, sonlu ve sonsuz toplamlar için formüller içerir. Toplamları değerlendirmek için diğer araçlarla birlikte kullanılabilir.
Bkz. Faulhaber formülü.
![{\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e82797674c101a71a773fa28db688ccaba2e827)
İlk birkaç değer şunlardır:
![{\displaystyle \sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/615f66562931b8bfd0238dc8ccc87b7a6e83d9e8)
![{\displaystyle \sum _{k=1}^{m}k^{2}={\frac {m(m+1)(2m+1)}{6}}={\frac {m^{3}}{3}}+{\frac {m^{2}}{2}}+{\frac {m}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590a25a336ef2d10df6962aee36d70dc8c623a5f)
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83655857c974dd27c9b29de8cda04d7c65d334e3)
Bkz. zeta sabitleri.
![{\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39c16e56068bfb1b7c7a16876faecbd23cae1fb9)
İlk birkaç değer şunlardır:
(Basel problemi) ![{\displaystyle \zeta (4)=\sum _{k=1}^{\infty }{\frac {1}{k^{4}}}={\frac {\pi ^{4}}{90}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57d340ce3e07c8d682543de1ee543ddb28dbf071)
![{\displaystyle \zeta (6)=\sum _{k=1}^{\infty }{\frac {1}{k^{6}}}={\frac {\pi ^{6}}{945}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c150edab196b63b262f0bcbb971ee895456f8e4)
Sonlu toplamlar:
, (geometrik seri) ![{\displaystyle \sum _{k=0}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/791932dcfd867eb4be8b21d9777dc8a9fd808553)
![{\displaystyle \sum _{k=1}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}-1={\frac {z-z^{n+1}}{1-z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbd311a88aea26c742eac8b12d2d64fca95cdf7)
![{\displaystyle \sum _{k=1}^{n}kz^{k}=z{\frac {1-(n+1)z^{n}+nz^{n+1}}{(1-z)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba5195ab25644b0202fb60e7c30b94d044ea38d)
![{\displaystyle \sum _{k=1}^{n}k^{2}z^{k}=z{\frac {1+z-(n+1)^{2}z^{n}+(2n^{2}+2n-1)z^{n+1}-n^{2}z^{n+2}}{(1-z)^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5274ec4b72fcd2bb8ed27ddf604ed21d8dd126f2)
![{\displaystyle \sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a59ad2bafdc84f1a2ed59d06acdf45a9cb4789)
Sonsuz toplamlar,
için geçerli (bkz. polilogaritma):
![{\displaystyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/269bc4ebc751699b90632451c1506b0d12aef7a9)
Aşağıdaki, düşük tam sayı mertebeli polilogaritmaları kapalı form içinde özyinelemeli olarak hesaplamak için yararlı bir özelliktir:
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {Li} _{n}(z)={\frac {\operatorname {Li} _{n-1}(z)}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/351a637191549347b91528e95bbf2be037723670)
![{\displaystyle \operatorname {Li} _{1}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}=-\ln(1-z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78c0907fa4e026586a3dec2121860a12c13a62c5)
![{\displaystyle \operatorname {Li} _{0}(z)=\sum _{k=1}^{\infty }z^{k}={\frac {z}{1-z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5a61f7feaffd247a5450eba4968debd0f9bf6e)
![{\displaystyle \operatorname {Li} _{-1}(z)=\sum _{k=1}^{\infty }kz^{k}={\frac {z}{(1-z)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2505cfc24d99fe2c95e297738310c1347577f017)
![{\displaystyle \operatorname {Li} _{-2}(z)=\sum _{k=1}^{\infty }k^{2}z^{k}={\frac {z(1+z)}{(1-z)^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d703061c9125105bede161bf3adc41091b2fb830)
![{\displaystyle \operatorname {Li} _{-3}(z)=\sum _{k=1}^{\infty }k^{3}z^{k}={\frac {z(1+4z+z^{2})}{(1-z)^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c15985776b2b6a3638ec04c0bf292b81cd6b72a)
![{\displaystyle \operatorname {Li} _{-4}(z)=\sum _{k=1}^{\infty }k^{4}z^{k}={\frac {z(1+z)(1+10z+z^{2})}{(1-z)^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f08ae7cc5ef199773da7054d9ba3b27aec21012d)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3c8535bc3feb0e123e11fe343171dd9d4776da)
(bkz. Poisson dağılımı ortalaması)
(bkz. Poisson dağılımının ikinci momenti) ![{\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62129fb023e2b6de038703c670c0394abdb87315)
![{\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738269671a82829e80dca30df6a8c4aa93c98653)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff42a20c13815fd8611f979983110d5f8d9b3a6)
burada;
Touchard polinomlarıdır.
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eeb6209d2ef99d44eb022f43b79787eade4c648)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)!}}=\sinh z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eed9faf752bff168c51a2901e44421778e377b6)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9386a3bfce6368adbad6c7962f37b18b9b995012)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k}}{(2k)!}}=\cosh z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e495ed1e2d351c9644a9b2b9b62814f0255d911)
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2256f274843b5a8dd7338fcd46d89457f27d39b8)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tanh z,|z|<{\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b10f67088d6d4a62eee48692deda3065a9ef72f8)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\cot z,|z|<\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/462f64ebe4b22d9eb36d69972a2c16259d72ea16)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\coth z,|z|<\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/00bfdc23630f34df2a588dcd3f1d5c7b3c9fc6f5)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\csc z,|z|<\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d223384181921eadadcc9acb38bbbd886d85c7ee)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\operatorname {csch} z,|z|<\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7564ad5932fa5f7084599d879730a4935370aab)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}E_{2k}z^{2k}}{(2k)!}}=\operatorname {sech} z,|z|<{\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b593907398cd4d3d157e0d4893ffe184fb1c9c67)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ea5a9b6c4c1072ff899840964d463dc890e1f6)
(versine)
[1] (haversine) ![{\displaystyle \sum _{k=0}^{\infty }{\frac {(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\arcsin z,|z|\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc3700c4addbf8311c6ff90b93ac759a750d6d8)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\operatorname {arcsinh} {z},|z|\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e915cadf00a2f6f95ccc6ae99dbf5c5b574a820b)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2k+1}}=\arctan z,|z|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bde385b223a3706eb46a282d932a6dc758bbd8fa)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{2k+1}}=\operatorname {arctanh} z,|z|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33cab9855e7ab0d8b6e59cdfe1e8e99cef53d093)
![{\displaystyle \ln 2+\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|\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea418d43688db9537a8b965838306a48a90840a7)
[2]
[2] ![{\displaystyle \sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \arcsin {z}},|z|\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7690094e2c29c30c517059014511d42f93f0912a)
(bkz Binom teoremi § Genelleştirilmiş Newton binom teoremi) - [3]
![{\displaystyle \sum _{k=0}^{\infty }{{\alpha +k-1} \choose k}z^{k}={\frac {1}{(1-z)^{\alpha }}},|z|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d69e6455c13c71f8e74ce0760ccc2f9fc11ac70d)
- [3]
, Catalan sayıları üreteç fonksiyonu - [3]
, Merkezi binom katsayıları üreteç fonksiyonu - [3]
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10c3c2d66060add977823b4848d7212af4b4b68f)
(Bkz harmonik sayılar, kendileri
olarak tanımlanmıştır)
![{\displaystyle \sum _{k=1}^{\infty }H_{k}z^{k}={\frac {-\ln(1-z)}{1-z}},|z|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/890b6859948e31ec717858a6a6b1582db3673345)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {H_{k}}{k+1}}z^{k+1}={\frac {1}{2}}\left[\ln(1-z)\right]^{2},\qquad |z|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c2c3f140738f0c5c61f88f041f311fbda3a340)
[2]
[2]
![{\displaystyle \sum _{k=0}^{n}{n \choose k}=2^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b30fdd28895f157a1d1f254f931879606064ce1c)
![{\displaystyle \sum _{k=0}^{n}(-1)^{k}{n \choose k}=0,{\text{ burada; }}n\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5457ed366d354eed13749b74f6b7d4ed51aaa7b5)
![{\displaystyle \sum _{k=0}^{n}{k \choose m}={n+1 \choose m+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fad96c9dbb6c1228a1f7264d6feea813478e34ea)
(bkz Çoklu küme)
(bkz Vandermonde özdeşliği)
Sinüsler ve kosinüsler toplamı, Fourier serileri'nde ortaya çıkar.
![{\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta )=-\ln \left(2\sin {\frac {\theta }{2}}\right),0<\theta <2\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e108de7fc84659c626acce1a6ed8e8c7403372)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e191794b1821b1f4608a4d21721396e2a705050b)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}\cos(k\theta )={\frac {1}{2}}\ln(2+2\cos \theta )=\ln \left(2\cos {\frac {\theta }{2}}\right),0\leq \theta <\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/86d3efde143fd7c2362f6a1e9901ddd660929bc5)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}\sin(k\theta )={\frac {\theta }{2}},-{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34f0bb1e3910c3dfb9f5c623390e1ed52eb73187)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(2k\theta )}{2k}}=-{\frac {1}{2}}\ln(2\sin \theta ),0<\theta <\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/220b903edbb592bbb3c4a60a3d10b018523064a6)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(2k\theta )}{2k}}={\frac {\pi -2\theta }{4}},0<\theta <\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2f32952d687fdbd4611dc18d6ac6fd4c217129)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {\cos[(2k+1)\theta ]}{2k+1}}={\frac {1}{2}}\ln \left(\cot {\frac {\theta }{2}}\right),0<\theta <\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1991f46f491715b581a7037b4125c14fe65025c)
,[4] ![{\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}=\pi \left({\dfrac {1}{2}}-\{x\}\right),\ x\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c1808477aa256c6d3673bafb5038444cfa5a43)
![{\displaystyle \sum \limits _{k=1}^{\infty }{\frac {\sin \left(2\pi kx\right)}{k^{2n-1}}}=(-1)^{n}{\frac {(2\pi )^{2n-1}}{2(2n-1)!}}B_{2n-1}(\{x\}),\ x\in \mathbb {R} ,\ n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/668cd8d4f3d7ed4f11a0c53a480ee0a2997f38ca)
![{\displaystyle \sum \limits _{k=1}^{\infty }{\frac {\cos \left(2\pi kx\right)}{k^{2n}}}=(-1)^{n-1}{\frac {(2\pi )^{2n}}{2(2n)!}}B_{2n}(\{x\}),\ x\in \mathbb {R} ,\ n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/85541ae24cd3f1eefd81d183ddb66384e9139a7d)
[5] ![{\displaystyle \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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c9a71d157f3e6aecf7c679c9d826cf2ed78772)
![{\displaystyle \sum _{k=0}^{n}\cos(\theta +k\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cos(\theta +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ece3ee92af0be40bcb51db92ab4286a96a49064d)
![{\displaystyle \sum _{k=1}^{n-1}\sin {\frac {\pi k}{n}}=\cot {\frac {\pi }{2n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1e592cdc3214ad2a61e0a4d6c8c171b9bbc237)
![{\displaystyle \sum _{k=1}^{n-1}\sin {\frac {2\pi k}{n}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538dd88d3f15d24a398e3f106d0a6092725fbeca)
[6] ![{\displaystyle \sum _{k=1}^{n-1}\csc ^{2}{\frac {\pi k}{n}}={\frac {n^{2}-1}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/036c3d6e188cf05baf35356bf314e236fb5a45ed)
![{\displaystyle \sum _{k=1}^{n-1}\csc ^{4}{\frac {\pi k}{n}}={\frac {n^{4}+10n^{2}-11}{45}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e969e8c1e28c457892ad6902866438f84193c32)
[7] ![{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{2}+a^{2}}}={\frac {1+a\pi \coth(a\pi )}{2a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1fc8f8afa2921f121e9d5b13b9c03a3b9f7dac)
![{\displaystyle \displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{4}+4a^{4}}}={\dfrac {1}{8a^{4}}}+{\dfrac {\pi (\sinh(2\pi a)+\sin(2\pi a))}{8a^{3}(\cosh(2\pi a)-\cos(2\pi a))}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34ea360b8b510486913cfdebaa4649472238e43b)
'nin herhangi bir rasyonel fonksiyon'unun sonsuz bir serisi, burada açıklandığı gibi kısmi kesirlere ayrıştırma[8] kullanılarak poligama fonksiyonu'nun sonlu bir serisine indirgenebilir. Bu gerçek, rasyonel fonksiyonların sonlu serilerine de uygulanabilir ve seri çok sayıda terim içerdiğinde bile sonucun sabit zamanda hesaplanmasına izin verir.
(bkz. Landsberg–Schaar bağıntısı) ![{\displaystyle \displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aee717a740629f569ad7c408608acb53f1ec4bd)
Bu numerik seriler, yukarıda listelenen serilerdeki sayılar eklenerek bulunabilir.
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots =\ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4ff44fdda82c3dd0f30099b4a1b820d8d298ae)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{2k-1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0456ed6c97eb961e591f1f0c761f4f661e4c63)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4fef2574bd4ee746eab71ebfd529bcaa59ef2c)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k)!}}={\frac {1}{0!}}+{\frac {1}{2!}}+{\frac {1}{4!}}+{\frac {1}{6!}}+{\frac {1}{8!}}+\cdots ={\frac {1}{2}}\left(e+{\frac {1}{e}}\right)=\cosh 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e5663f60730e245e68c186e7a31b68355746b6)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(3k)!}}={\frac {1}{0!}}+{\frac {1}{3!}}+{\frac {1}{6!}}+{\frac {1}{9!}}+{\frac {1}{12!}}+\cdots ={\frac {1}{3}}\left(e+{\frac {2}{\sqrt {e}}}\cos {\frac {\sqrt {3}}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/096a96043c26ef8f5a6637595a95b0f1e01741c2)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(4k)!}}={\frac {1}{0!}}+{\frac {1}{4!}}+{\frac {1}{8!}}+{\frac {1}{12!}}+{\frac {1}{16!}}+\cdots ={\frac {1}{2}}\left(\cos 1+\cosh 1\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1948bb61147cdb0c9e5bcf8b8be651dfa6be979)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}={\frac {1}{1!}}-{\frac {1}{3!}}+{\frac {1}{5!}}-{\frac {1}{7!}}+{\frac {1}{9!}}+\cdots =\sin 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11af20e57510a33a32090b23c25d8bb494ad2010)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}={\frac {1}{0!}}-{\frac {1}{2!}}+{\frac {1}{4!}}-{\frac {1}{6!}}+{\frac {1}{8!}}+\cdots =\cos 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46799aba48900678fd422c69af967081fd488be8)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}+1}}={\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{10}}+{\frac {1}{17}}+\cdots ={\frac {1}{2}}(\pi \coth \pi -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be42ffe1f589602a8f98859496f18d58e15ed483)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}+1}}=-{\frac {1}{2}}+{\frac {1}{5}}-{\frac {1}{10}}+{\frac {1}{17}}+\cdots ={\frac {1}{2}}(\pi \operatorname {csch} \pi -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d31bc0f119aef2f337ce1e76c75e9611114d27b)
![{\displaystyle 3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots =\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f85b809900c2b876145b6bb606839a9e1511e07)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{T_{k}}}={\frac {1}{1}}+{\frac {1}{3}}+{\frac {1}{6}}+{\frac {1}{10}}+{\frac {1}{15}}+\cdots =2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5adb4a28bbbf96d112cc159b99f4dc4752ea293a)
Burada;
![{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{Te_{k}}}={\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{20}}+{\frac {1}{35}}+\cdots ={\frac {3}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7adf22bfb6785d1a675c2e4ad7488af1de11758c)
Burada;
![{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}={\frac {1}{1\times 2}}+{\frac {1}{3\times 4}}+{\frac {1}{5\times 6}}+{\frac {1}{7\times 8}}+{\frac {1}{9\times 10}}+\cdots =\ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/940b1790cbab3f08e97e6bbb6559c627404b1806)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2^{k}k}}={\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{24}}+{\frac {1}{64}}+{\frac {1}{160}}+\cdots =\ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c738c3cb116898bbb8c511d8200e7f784553fcc7)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{2^{k}k}}+\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{3^{k}k}}={\Bigg (}{\frac {1}{2}}+{\frac {1}{3}}{\Bigg )}-{\Bigg (}{\frac {1}{8}}+{\frac {1}{18}}{\Bigg )}+{\Bigg (}{\frac {1}{24}}+{\frac {1}{81}}{\Bigg )}-{\Bigg (}{\frac {1}{64}}+{\frac {1}{324}}{\Bigg )}+\cdots =\ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0de6782322e0c4eafd43e5c77931a661be96ce)
![{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{3^{k}k}}+\sum _{k=1}^{\infty }{\frac {1}{4^{k}k}}={\Bigg (}{\frac {1}{3}}+{\frac {1}{4}}{\Bigg )}+{\Bigg (}{\frac {1}{18}}+{\frac {1}{32}}{\Bigg )}+{\Bigg (}{\frac {1}{81}}+{\frac {1}{192}}{\Bigg )}+{\Bigg (}{\frac {1}{324}}+{\frac {1}{1024}}{\Bigg )}+\cdots =\ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28941a5ef0568162c5c95c954589196ca92b40f8)
- ^ Weisstein, Eric W. "Haversine". MathWorld. Wolfram Research, Inc. 10 Mart 2005 tarihinde kaynağından arşivlendi. Erişim tarihi: 6 Kasım 2015.
- ^ a b c d Wilf, Herbert R. (1994). generatingfunctionology (PDF). Academic Press, Inc. 27 Nisan 2021 tarihinde kaynağından arşivlendi (PDF). Erişim tarihi: 13 Temmuz 2023.
- ^ a b c d "Theoretical computer science cheat sheet" (PDF). 10 Haziran 2003 tarihinde kaynağından (PDF) arşivlendi.
- ^
fonksiyonun Fourier açılımını
aralığında hesaplayın: ![{\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }c_{n}\sin[nx]+d_{n}\cos[nx]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5b6fd91cf5e77255955c2b09cdc203bcb5bf73)
- ^ "Bernoulli polynomials: Series representations (subsection 06/02)". Wolfram Research. 28 Eylül 2011 tarihinde kaynağından arşivlendi. Erişim tarihi: 2 Haziran 2011.
- ^ Hofbauer, Josef. "A simple proof of 1 + 1/22 + 1/32 + ··· = π2/6 and related identities" (PDF). 20 Temmuz 2007 tarihinde kaynağından (PDF) arşivlendi. Erişim tarihi: 2 Haziran 2011.
- ^ Sondow, Jonathan; Weisstein, Eric W. "Riemann Zeta Function (eq. 52)". MathWorld—A Wolfram Web Resource. 17 Ağustos 2000 tarihinde kaynağından arşivlendi.
- ^ Abramowitz, Milton; Stegun, Irene (1964). "6.4 Polygamma functions". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. s. 260. ISBN 0-486-61272-4.
- İntegraller listesi içeren birçok kitapta, seriler listesi de vardır.
![{\displaystyle {\frac {\pi }{4}}=\sum _{k=0}^{\infty }{\frac {\left(-1\right)^{k}}{2k+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed1d73c14fa9b4617c9469800bf02645cfe1a01)