Artículo de la lista de Wikipedia
Esta lista de series matemáticas contiene fórmulas para sumas finitas e infinitas. Se puede utilizar junto con otras herramientas para evaluar sumas.
Sumas de poderes [ editar ] Vea la fórmula de Faulhaber .
∑ k = 0 metro k norte - 1 = B norte ( metro + 1 ) - B norte norte {\ Displaystyle \ sum _ {k = 0} ^ {m} k ^ {n-1} = {\ frac {B_ {n} (m + 1) -B_ {n}} {n}}} Los primeros valores son:
∑ k = 1 metro k = metro ( metro + 1 ) 2 {\ Displaystyle \ sum _ {k = 1} ^ {m} k = {\ frac {m (m + 1)} {2}}} ∑ k = 1 metro k 2 = metro ( metro + 1 ) ( 2 metro + 1 ) 6 = metro 3 3 + metro 2 2 + metro 6 {\ 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}}} ∑ k = 1 metro k 3 = [ metro ( metro + 1 ) 2 ] 2 = metro 4 4 + metro 3 2 + metro 2 4 {\ 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}}} Ver constantes zeta .
ζ ( 2 norte ) = ∑ k = 1 ∞ 1 k 2 norte = ( - 1 ) norte + 1 B 2 norte ( 2 π ) 2 norte 2 ( 2 norte ) ! {\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}} Los primeros valores son:
ζ ( 2 ) = ∑ k = 1 ∞ 1 k 2 = π 2 6 {\displaystyle \zeta (2)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} problema de Basilea ) ζ ( 4 ) = ∑ k = 1 ∞ 1 k 4 = π 4 90 {\displaystyle \zeta (4)=\sum _{k=1}^{\infty }{\frac {1}{k^{4}}}={\frac {\pi ^{4}}{90}}} ζ ( 6 ) = ∑ k = 1 ∞ 1 k 6 = π 6 945 {\displaystyle \zeta (6)=\sum _{k=1}^{\infty }{\frac {1}{k^{6}}}={\frac {\pi ^{6}}{945}}} Power series [ editar ] Polilogaritmos de orden inferior [ editar ] Sumas finitas:
∑ k = m n z k = z m − z n + 1 1 − z {\displaystyle \sum _{k=m}^{n}z^{k}={\frac {z^{m}-z^{n+1}}{1-z}}} serie geométrica ) ∑ k = 0 n z k = 1 − z n + 1 1 − z {\displaystyle \sum _{k=0}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}} ∑ k = 1 n z k = 1 − z n + 1 1 − z − 1 = z − z n + 1 1 − z {\displaystyle \sum _{k=1}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}-1={\frac {z-z^{n+1}}{1-z}}} ∑ k = 1 n k z k = z 1 − ( n + 1 ) z n + n z n + 1 ( 1 − z ) 2 {\displaystyle \sum _{k=1}^{n}kz^{k}=z{\frac {1-(n+1)z^{n}+nz^{n+1}}{(1-z)^{2}}}} ∑ k = 1 n k 2 z k = z 1 + z − ( n + 1 ) 2 z n + ( 2 n 2 + 2 n − 1 ) z n + 1 − n 2 z n + 2 ( 1 − z ) 3 {\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}}}} ∑ k = 1 n k m z k = ( z d d z ) m 1 − z n + 1 1 − z {\displaystyle \sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}} Sumas infinitas, válidas para (ver polilogaritmo ): | z | < 1 {\displaystyle |z|<1}
Li n ( z ) = ∑ k = 1 ∞ z k k n {\displaystyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}} La siguiente es una propiedad útil para calcular polilogaritmos de orden entero bajo de forma recursiva en forma cerrada :
d d z Li n ( z ) = Li n − 1 ( z ) z {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {Li} _{n}(z)={\frac {\operatorname {Li} _{n-1}(z)}{z}}} Li 1 ( z ) = ∑ k = 1 ∞ z k k = − ln ( 1 − z ) {\displaystyle \operatorname {Li} _{1}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}=-\ln(1-z)} Li 0 ( z ) = ∑ k = 1 ∞ z k = z 1 − z {\displaystyle \operatorname {Li} _{0}(z)=\sum _{k=1}^{\infty }z^{k}={\frac {z}{1-z}}} Li − 1 ( z ) = ∑ k = 1 ∞ k z k = z ( 1 − z ) 2 {\displaystyle \operatorname {Li} _{-1}(z)=\sum _{k=1}^{\infty }kz^{k}={\frac {z}{(1-z)^{2}}}} Li − 2 ( z ) = ∑ k = 1 ∞ k 2 z k = z ( 1 + z ) ( 1 − z ) 3 {\displaystyle \operatorname {Li} _{-2}(z)=\sum _{k=1}^{\infty }k^{2}z^{k}={\frac {z(1+z)}{(1-z)^{3}}}} Li − 3 ( z ) = ∑ k = 1 ∞ k 3 z k = z ( 1 + 4 z + z 2 ) ( 1 − z ) 4 {\displaystyle \operatorname {Li} _{-3}(z)=\sum _{k=1}^{\infty }k^{3}z^{k}={\frac {z(1+4z+z^{2})}{(1-z)^{4}}}} Li − 4 ( z ) = ∑ k = 1 ∞ k 4 z k = z ( 1 + z ) ( 1 + 10 z + z 2 ) ( 1 − z ) 5 {\displaystyle \operatorname {Li} _{-4}(z)=\sum _{k=1}^{\infty }k^{4}z^{k}={\frac {z(1+z)(1+10z+z^{2})}{(1-z)^{5}}}} Función exponencial [ editar ] ∑ k = 0 ∞ z k k ! = e z {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}} ∑ k = 0 ∞ k z k k ! = z e z {\displaystyle \sum _{k=0}^{\infty }k{\frac {z^{k}}{k!}}=ze^{z}} la distribución de Poisson ) ∑ k = 0 ∞ k 2 z k k ! = ( z + z 2 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}} segundo momento de la distribución de Poisson) ∑ k = 0 ∞ k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}} ∑ k = 0 ∞ k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}} ∑ k = 0 ∞ k n z k k ! = z d d z ∑ k = 0 ∞ k n − 1 z k k ! = e z T n ( z ) {\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)} donde están los polinomios de Touchard . T n ( z ) {\displaystyle T_{n}(z)}
Relación de funciones trigonométricas, trigonométricas inversas, hiperbólicas e hiperbólicas inversas [ editar ] ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin z {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z} ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) ! = sinh z {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)!}}=\sinh z} ∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k ) ! = cos z {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z} ∑ k = 0 ∞ z 2 k ( 2 k ) ! = cosh z {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k}}{(2k)!}}=\cosh z} ∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 2 k − 1 ) 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = tan z , | z | < π 2 {\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}}} ∑ k = 1 ∞ ( 2 2 k − 1 ) 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = tanh z , | z | < π 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tanh z,|z|<{\frac {\pi }{2}}} ∑ k = 0 ∞ ( − 1 ) k 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = cot z , | z | < π {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\cot z,|z|<\pi } ∑ k = 0 ∞ 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = coth z , | z | < π {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\coth z,|z|<\pi } ∑ k = 0 ∞ ( − 1 ) k − 1 ( 2 2 k − 2 ) B 2 k z 2 k − 1 ( 2 k ) ! = csc z , | z | < π {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\csc z,|z|<\pi } ∑ k = 0 ∞ − ( 2 2 k − 2 ) B 2 k z 2 k − 1 ( 2 k ) ! = csch z , | z | < π {\displaystyle \sum _{k=0}^{\infty }{\frac {-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\operatorname {csch} z,|z|<\pi } ∑ k = 0 ∞ ( − 1 ) k E 2 k z 2 k ( 2 k ) ! = sech z , | z | < π 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}E_{2k}z^{2k}}{(2k)!}}=\operatorname {sech} z,|z|<{\frac {\pi }{2}}} ∑ k = 0 ∞ E 2 k z 2 k ( 2 k ) ! = sec z , | z | < π 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}} ∑ k = 1 ∞ ( − 1 ) k − 1 z 2 k ( 2 k ) ! = ver z {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}=\operatorname {ver} z} versine ) ∑ k = 1 ∞ ( − 1 ) k − 1 z 2 k 2 ( 2 k ) ! = hav z {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}=\operatorname {hav} z} [1] ∑ k = 0 ∞ ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsin z , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\arcsin z,|z|\leq 1} ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsinh z , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\operatorname {arcsinh} {z},|z|\leq 1} ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 2 k + 1 = arctan z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2k+1}}=\arctan z,|z|<1} ∑ k = 0 ∞ z 2 k + 1 2 k + 1 = arctanh z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {z^{2k+1}}{2k+1}}=\operatorname {arctanh} z,|z|<1} ln 2 + ∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 k ) ! z 2 k 2 2 k + 1 k ( k ! ) 2 = ln ( 1 + 1 + z 2 ) , | z | ≤ 1 {\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} Denominadores factoriales modificados [ editar ] ∑ k = 0 ∞ ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 − 1 − z z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1} [2] ∑ k = 0 ∞ 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin z ) 2 , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\arcsin {z}\right)^{2},|z|\leq 1} [2] ∑ n = 0 ∞ ∏ k = 0 n − 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + ∑ n = 0 ∞ α ∏ k = 0 n − 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin z , | z | ≤ 1 {\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} Coeficientes binomiales [ editar ] ( 1 + z ) α = ∑ k = 0 ∞ ( α k ) z k , | z | < 1 {\displaystyle (1+z)^{\alpha }=\sum _{k=0}^{\infty }{\alpha \choose k}z^{k},|z|<1} [3] ∑ k = 0 ∞ ( α + k − 1 k ) z k = 1 ( 1 − z ) α , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{{\alpha +k-1} \choose k}z^{k}={\frac {1}{(1-z)^{\alpha }}},|z|<1} [3] , función generadora de los números catalanes ∑ k = 0 ∞ 1 k + 1 ( 2 k k ) z k = 1 − 1 − 4 z 2 z , | z | ≤ 1 4 {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k+1}}{2k \choose k}z^{k}={\frac {1-{\sqrt {1-4z}}}{2z}},|z|\leq {\frac {1}{4}}} [3] , función generadora de los coeficientes binomiales centrales ∑ k = 0 ∞ ( 2 k k ) z k = 1 1 − 4 z , | z | < 1 4 {\displaystyle \sum _{k=0}^{\infty }{2k \choose k}z^{k}={\frac {1}{\sqrt {1-4z}}},|z|<{\frac {1}{4}}} [3] ∑ k = 0 ∞ ( 2 k + α k ) z k = 1 1 − 4 z ( 1 − 1 − 4 z 2 z ) α , | z | < 1 4 {\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}}} Números armónicos [ editar ] (Ver números armónicos , ellos mismos definidos ) H n = ∑ j = 1 n 1 j {\textstyle H_{n}=\sum _{j=1}^{n}{\frac {1}{j}}}
∑ k = 1 ∞ H k z k = − ln ( 1 − z ) 1 − z , | z | < 1 {\displaystyle \sum _{k=1}^{\infty }H_{k}z^{k}={\frac {-\ln(1-z)}{1-z}},|z|<1} ∑ k = 1 ∞ H k k + 1 z k + 1 = 1 2 [ ln ( 1 − z ) ] 2 , | z | < 1 {\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} ∑ k = 1 ∞ ( − 1 ) k − 1 H 2 k 2 k + 1 z 2 k + 1 = 1 2 arctan z log ( 1 + z 2 ) , | z | < 1 {\displaystyle \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} [2] ∑ n = 0 ∞ ∑ k = 0 2 n ( − 1 ) k 2 k + 1 z 4 n + 2 4 n + 2 = 1 4 arctan z log 1 + z 1 − z , | z | < 1 {\displaystyle \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} [2] Coeficientes binomiales [ editar ] ∑ k = 0 n ( n k ) = 2 n {\displaystyle \sum _{k=0}^{n}{n \choose k}=2^{n}} ∑ k = 0 n ( − 1 ) k ( n k ) = 0 , where n > 0 {\displaystyle \sum _{k=0}^{n}(-1)^{k}{n \choose k}=0,{\text{ where }}n>0} ∑ k = 0 n ( k m ) = ( n + 1 m + 1 ) {\displaystyle \sum _{k=0}^{n}{k \choose m}={n+1 \choose m+1}} ∑ k = 0 n ( m + k − 1 k ) = ( n + m n ) {\displaystyle \sum _{k=0}^{n}{m+k-1 \choose k}={n+m \choose n}} ∑ k = 0 n ( α k ) ( β n − k ) = ( α + β n ) {\displaystyle \sum _{k=0}^{n}{\alpha \choose k}{\beta \choose n-k}={\alpha +\beta \choose n}} Funciones trigonométricas [ editar ] Las sumas de senos y cosenos surgen en series de Fourier .
∑ k = 1 ∞ sin ( k θ ) k = π − θ 2 , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi } ∑ k = 1 ∞ cos ( k θ ) k = − 1 2 ln ( 2 − 2 cos θ ) , θ ∈ R {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta ),\theta \in \mathbb {R} } ∑ k = 0 ∞ sin [ ( 2 k + 1 ) θ ] 2 k + 1 = π 4 , 0 < θ < π {\displaystyle \sum _{k=0}^{\infty }{\frac {\sin[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi } [4] B n ( x ) = − n ! 2 n − 1 π n ∑ k = 1 ∞ 1 k n cos ( 2 π k x − π n 2 ) , 0 < x < 1 {\displaystyle 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} [5] ∑ k = 0 n sin ( θ + k α ) = sin ( n + 1 ) α 2 sin ( θ + n α 2 ) sin α 2 {\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}}}}} ∑ k = 0 n cos ( θ + k α ) = sin ( n + 1 ) α 2 cos ( θ + n α 2 ) sin α 2 {\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}}}}} ∑ k = 1 n − 1 sin π k n = cot π 2 n {\displaystyle \sum _{k=1}^{n-1}\sin {\frac {\pi k}{n}}=\cot {\frac {\pi }{2n}}} ∑ k = 1 n − 1 sin 2 π k n = 0 {\displaystyle \sum _{k=1}^{n-1}\sin {\frac {2\pi k}{n}}=0} ∑ k = 0 n − 1 csc 2 ( θ + π k n ) = n 2 csc 2 ( n θ ) {\displaystyle \sum _{k=0}^{n-1}\csc ^{2}\left(\theta +{\frac {\pi k}{n}}\right)=n^{2}\csc ^{2}(n\theta )} [6] ∑ k = 1 n − 1 csc 2 π k n = n 2 − 1 3 {\displaystyle \sum _{k=1}^{n-1}\csc ^{2}{\frac {\pi k}{n}}={\frac {n^{2}-1}{3}}} ∑ k = 1 n − 1 csc 4 π k n = n 4 + 10 n 2 − 11 45 {\displaystyle \sum _{k=1}^{n-1}\csc ^{4}{\frac {\pi k}{n}}={\frac {n^{4}+10n^{2}-11}{45}}} Funciones racionales [ editar ] ∑ n = a + 1 ∞ a n 2 − a 2 = 1 2 H 2 a {\displaystyle \sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}} [7] ∑ n = 0 ∞ 1 n 2 + a 2 = 1 + a π coth ( a π ) 2 a 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{2}+a^{2}}}={\frac {1+a\pi \coth(a\pi )}{2a^{2}}}} ∑ n = 0 ∞ 1 n 4 + 4 a 4 = 1 8 a 4 + π ( sinh ( 2 π a ) + sin ( 2 π a ) ) 8 a 3 ( cosh ( 2 π a ) − cos ( 2 π a ) ) {\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))}}} Una serie infinita de cualquier función racional de puede reducirse a una serie finita de funciones poligamma , mediante el uso de descomposición de fracciones parciales . [8] Este hecho también se puede aplicar a series finitas de funciones racionales, lo que permite calcular el resultado en tiempo constante incluso cuando la serie contiene una gran cantidad de términos. n {\displaystyle n} Función exponencial [ editar ] 1 p ∑ n = 0 p − 1 exp ( 2 π i n 2 q p ) = e π i / 4 2 q ∑ n = 0 2 q − 1 exp ( − π i n 2 p 2 q ) {\displaystyle \displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)} ∑ n = − ∞ ∞ e − π n 2 = π 4 Γ ( 3 4 ) {\displaystyle \displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}} Ver también [ editar ] Serie (matemáticas) Lista de integrales Suma § Identidades Serie Taylor Teorema binomial Serie de Gregory Enciclopedia en línea de secuencias de enteros Notas [ editar ] ^ Weisstein, Eric W. "Haversine" . MathWorld . Wolfram Research, Inc. Archivado desde el original el 10 de marzo de 2005 . Consultado el 6 de noviembre de 2015 . ↑ a b c d Wilf, Herbert R. (1994). función de generación (PDF) . Prensa académica, Inc. ^ a b c d "Hoja de trucos de informática teórica" (PDF) . ^
Calcule la expansión de Fourier de la funciónen el intervalo: f ( x ) = π 4 {\displaystyle f(x)={\frac {\pi }{4}}} 0 < x < π {\displaystyle 0<x<\pi } π 4 = ∑ n = 0 ∞ c n sin [ n x ] + d n cos [ n x ] {\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }c_{n}\sin[nx]+d_{n}\cos[nx]} ⇒ { c n = { 1 n ( n odd ) 0 ( n even ) d n = 0 ( ∀ n ) {\displaystyle \Rightarrow {\begin{cases}c_{n}={\begin{cases}{\frac {1}{n}}\quad (n{\text{ odd}})\\0\quad (n{\text{ even}})\end{cases}}\\d_{n}=0\quad (\forall n)\end{cases}}} ^ "Polinomios de Bernoulli: representaciones en serie (subsección 02/06)" . Wolfram Research . Consultado el 2 de junio de 2011 . ^ Hofbauer, Josef. "Una prueba simple de 1 + 1/2 ^ 2 + 1/3 ^ 2 + ... = PI ^ 2/6 e identidades relacionadas" (PDF) . Consultado el 2 de junio de 2011 . ^ Sondow, Jonathan; Weisstein, Eric W. "Función Zeta de Riemann (ec. 52)" . MathWorld: un recurso web de Wolfram . ^ Abramowitz, Milton ; Stegun, Irene (1964). "6.4 Funciones de Polygamma" . Manual de funciones matemáticas con fórmulas, gráficos y tablas matemáticas . pag. 260 . ISBN Referencias [ editar ] Muchos libros con una lista de integrales también tienen una lista de series. 
![{\ 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)
![\ 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)
![\ 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)
![{\ Displaystyle \ sum _ {k = 0} ^ {\ infty} {\ frac {\ sin [(2k + 1) \ theta]} {2k + 1}} = {\ frac {\ pi} {4}} , 0 <\ theta <\ pi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf41e942b227c04e70af874e45bddb621602e58)
![{\ 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)
![{\ 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)
