CÁLCULO GRACELI SINTÉTICO [30]

Aqui, considera-se que $0^{0}$ vale $1$
$B_{n}(x)$ é um polinômio de Bernoulli.
$B_{n}$ é um número de Bernoulli, e aqui, $B_{1}=-{\frac {1}{2}}.$
$E_{n}$ é um número de Euler.
$\zeta (s)$ é a função zeta de Riemann.
$\Gamma (z)$ é a função gama.
$\psi _{n}(z)$ é uma função poligama.
$\operatorname {Li} _{s}(z)$ é um polilogaritmo .
$n \choose k$ é o coeficiente binomial
$\exp(x)$ denota a exponencial de $x$

Soma de potências[editar | editar código-fonte]

Ver a fórmula de Faulhaber.

$\sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}$

Os primeiros valores são:

$\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{m}k^{2}={\frac {m(m+1)(2m+1)}{6}}={\frac {m^{3}}{3}}+{\frac {m^{2}}{2}}+{\frac {m}{6}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{m}k^{3}=\left[{\frac {m(m+1)}{2}}\right]^{2}={\frac {m^{4}}{4}}+{\frac {m^{3}}{2}}+{\frac {m^{2}}{4}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Ver constantes zeta.

$\zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Os primeiros valores são:

$\zeta (2)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\zeta (4)=\sum _{k=1}^{\infty }{\frac {1}{k^{4}}}={\frac {\pi ^{4}}{90}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\zeta (6)=\sum _{k=1}^{\infty }{\frac {1}{k^{6}}}={\frac {\pi ^{6}}{945}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Séries de potências[editar | editar código-fonte]

Polilogaritmos de ordem baixa[editar | editar código-fonte]

Somas com uma quantidade finita de termos:

$\sum _{k=0}^{n}z^{k}={\frac {1-z^{n+1}}{1-z}}$ , $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n}kz^{k}=z{\frac {1-(n+1)z^{n}+nz^{n+1}}{(1-z)^{2}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n}k^{2}z^{k}=z{\frac {1+z-(n+1)^{2}z^{n}+(2n^{2}+2n-1)z^{n+1}-n^{2}z^{n+2}}{(1-z)^{3}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Somas com uma infinidade de termos, válidas para $|z|<1$ (ver polilogaritmo):

$\operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

A propriedade a seguir é útil para calcular polilogaritmos de ordem inteira baixa recursivamente de forma fechada:

${\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {Li} _{n}(z)={\frac {\operatorname {Li} _{n-1}(z)}{z}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\operatorname {Li} _{1}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}=-\ln(1-z)$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\operatorname {Li} _{0}(z)=\sum _{k=1}^{\infty }z^{k}={\frac {z}{1-z}}$
$\operatorname {Li} _{-1}(z)=\sum _{k=1}^{\infty }kz^{k}={\frac {z}{(1-z)^{2}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\operatorname {Li} _{-2}(z)=\sum _{k=1}^{\infty }k^{2}z^{k}={\frac {z(1+z)}{(1-z)^{3}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\operatorname {Li} _{-3}(z)=\sum _{k=1}^{\infty }k^{3}z^{k}={\frac {z(1+4z+z^{2})}{(1-z)^{4}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\operatorname {Li} _{-4}(z)=\sum _{k=1}^{\infty }k^{4}z^{k}={\frac {z(1+z)(1+10z+z^{2})}{(1-z)^{5}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Função exponencial[editar | editar código-fonte]

$\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }k{\frac {z^{k}}{k!}}=ze^{z}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

em que $T_{n}(z)$ são os polinômios de Touchard.

Funções trigonométricas, trigonométricas inversas, hiperbólicas e hiperbólicas inversas[editar | editar código-fonte]

$\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\mathrm {sen} \,z$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)!}}=\mathrm {senh} \,z$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k)!}}=\cos z$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {z^{2k}}{(2k)!}}=\cosh z$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tan z,|z|<{\frac {\pi }{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\tanh z,|z|<{\frac {\pi }{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\cot z,|z|<\pi$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {2^{2k}B_{2k}z^{2k-1}}{(2k)!}}=\coth z,|z|<\pi$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\csc z,|z|<\pi$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}}=\operatorname {csch} z,|z|<\pi$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}E_{2k}z^{2k}}{(2k)!}}=\operatorname {sech} z,|z|<{\frac {\pi }{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}=\operatorname {ver} z$ ( $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}=\operatorname {hav} z$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\mathrm {arcsen} \,z,|z|\leq 1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}}=\operatorname {arcsenh} {z},|z|\leq 1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2k+1}}=\arctan z,|z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{2k+1}}=\operatorname {arctanh} z,|z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\ln 2+\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^{2}}}=\ln \left(1+{\sqrt {1+z^{2}}}\right),|z|\leq 1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Denominadores fatoriais modificados[editar | editar código-fonte]

$\sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1$ ^[2] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\mathrm {arcsen} \,{z}\right)^{2},|z|\leq 1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \mathrm {arcsen} \,{z}},|z|\leq 1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Coeficientes binomiais[editar | editar código-fonte]

$(1+z)^{\alpha }=\sum _{k=0}^{\infty }{\alpha \choose k}z^{k},|z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
^[3] $\sum _{k=0}^{\infty }{{\alpha +k-1} \choose k}z^{k}={\frac {1}{(1-z)^{\alpha }}},|z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {1}{k+1}}{2k \choose k}z^{k}={\frac {1-{\sqrt {1-4z}}}{2z}},|z|\leq {\frac {1}{4}}$ , $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{2k \choose k}z^{k}={\frac {1}{\sqrt {1-4z}}},|z|<{\frac {1}{4}}$ , $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{2k+\alpha \choose k}z^{k}={\frac {1}{\sqrt {1-4z}}}\left({\frac {1-{\sqrt {1-4z}}}{2z}}\right)^{\alpha },|z|<{\frac {1}{4}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Números harmônicos[editar | editar código-fonte]

(Ver números harmônicos, que são definidos por ${\textstyle H_{n}=\sum _{j=1}^{n}{\frac {1}{j}}}$ ) $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

$\sum _{k=1}^{\infty }H_{k}z^{k}={\frac {-\ln(1-z)}{1-z}},|z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {H_{k}}{k+1}}z^{k+1}={\frac {1}{2}}\left[\ln(1-z)\right]^{2},\qquad |z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}H_{2k}}{2k+1}}z^{2k+1}={\frac {1}{2}}\arctan {z}\log {(1+z^{2})},\qquad |z|<1$ ^[2] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{n=0}^{\infty }\sum _{k=0}^{2n}{\frac {(-1)^{k}}{2k+1}}{\frac {z^{4n+2}}{4n+2}}={\frac {1}{4}}\arctan {z}\log {\frac {1+z}{1-z}},\qquad |z|<1$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Coeficientes binomiais[editar | editar código-fonte]

$\sum _{k=0}^{n}{n \choose k}=2^{n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}(-1)^{k}{n \choose k}=0,{\text{ where }}n>0$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}{k \choose m}={n+1 \choose m+1}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}{m+k-1 \choose k}={n+m \choose n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}{\alpha \choose k}{\beta \choose n-k}={\alpha +\beta \choose n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Funções trigonométricas[editar | editar código-fonte]

Soma de senos e cossenos surgem nas séries de Fourier.

$\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta ),\theta \in \mathbb {R}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{\infty }{\frac {\mathrm {sen} \,[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi$ ,^[4] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$B_{n}(x)=-{\frac {n!}{2^{n-1}\pi ^{n}}}\sum _{k=1}^{\infty }{\frac {1}{k^{n}}}\cos \left(2\pi kx-{\frac {\pi n}{2}}\right),0<x<1$ ^[5] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}\mathrm {sen} \,(\theta +k\alpha )={\frac {\mathrm {sen} \,{\frac {(n+1)\alpha }{2}}\mathrm {sen} \,(\theta +{\frac {n\alpha }{2}})}{\mathrm {sen} \,{\frac {\alpha }{2}}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n}\cos(\theta +k\alpha )={\frac {\mathrm {sen} \,{\frac {(n+1)\alpha }{2}}\cos(\theta +{\frac {n\alpha }{2}})}{\mathrm {sen} \,{\frac {\alpha }{2}}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n-1}\mathrm {sen} \,{\frac {\pi k}{n}}=\cot {\frac {\pi }{2n}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n-1}\mathrm {sen} \,{\frac {2\pi k}{n}}=0$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=0}^{n-1}\csc ^{2}\left(\theta +{\frac {\pi k}{n}}\right)=n^{2}\csc ^{2}(n\theta )$ ^[6] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n-1}\csc ^{2}{\frac {\pi k}{n}}={\frac {n^{2}-1}{3}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{k=1}^{n-1}\csc ^{4}{\frac {\pi k}{n}}={\frac {n^{4}+10n^{2}-11}{45}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Funções racionais[editar | editar código-fonte]

$\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ ^[7] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\sum _{n=0}^{\infty }{\frac {1}{n^{2}+a^{2}}}={\frac {1+a\pi \coth(a\pi )}{2a^{2}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
$\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{4}+4a^{4}}}={\dfrac {1+a\pi \coth(a\pi )}{8a^{4}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$
Uma série infinita de qualquer função racional de $n$ pode ser reduzida a uma série finita de funções poligama, pelo uso da decomposição em frações parciais.^[8] Esse fato também pode ser aplicado a séries finitas de funções racionais, permitindo que o resultado seja calculado em tempo constante, mesmo quando a série contém um grande número de termos.

Função exponencial[editar | editar código-fonte]

$\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$
$\displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

${\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i},$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

onde $\{x_{i}\}_{i=1}^{n}$ é um dada sequência de $n$ números.^[3]

Algumas propriedades[editar | editar código-fonte]

Sejam $\{x_{k}\}_{k\in \mathbb {N} },$ $\{y_{k}\}_{k\in \mathbb {N} }$ sequências (por exemplo, de números reais) e $\alpha$ um escalar. Então, temos:

1. $\sum _{i=m}^{n}\alpha x_{i}=\alpha \sum _{i=m}^{n}x_{i}$ ^[3] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

2. $\sum _{i=m}^{n}(x_{i}\pm y_{i})=\sum _{i=m}^{n}x_{i}\pm \sum _{i=m}^{n}y_{i}$ ^[3] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

3. $\sum _{i=m}^{m}x_{i}=x_{m}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

4. $\sum _{i=m}^{n}x_{i}=\sum _{i=m}^{p}x_{i}+\sum _{i=p+1}^{n}x_{i},\quad \forall m\leq p\leq n$ ^[3] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

5. $\sum _{i=m}^{n}x_{i}=\sum _{i=m+p}^{n+p}x_{i-p}$ ^[3] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

6. $\sum _{i=m}^{n}(x_{i+1}-x_{i})=x_{n+1}-x_{m}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

7. $\sum _{i=m}^{n}\sum _{j=k}^{l}x_{i}y_{j}=\sum _{i=m}^{n}x_{i}\sum _{j=k}^{l}y_{j}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

8. $\left|\sum _{i=m}^{n}x_{i}\right|\leq \sum _{i=m}^{n}|x_{i}|$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

9. $\sum _{n=0}^{t}x_{2n}+\sum _{n=0}^{t}x_{2n+1}=\sum _{n=0}^{2t+1}x_{n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

10. $\sum _{n=0}^{t}\sum _{i=0}^{z-1}x_{z\cdot n+i}=\sum _{n=0}^{z\cdot t+z-1}x_{n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

11. $\left(\sum _{k=0}^{n}a_{k}\right)\cdot \left(\sum _{k=0}^{n}b_{k}\right)=\sum _{k=0}^{2n}\sum _{i=0}^{k}a_{i}b_{k-i}-\sum _{k=0}^{n-1}\left(a_{k}\sum _{i=n+1}^{2n-k}b_{i}+b_{k}\sum _{i=n+1}^{2n-k}a_{i}\right)$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Nas propriedades acima, assumimos que as sequências $\{x_{k}\}_{k\in \mathbb {N} },$ $\{y_{k}\}_{k\in \mathbb {N} }$ pertencem a um espaço vetorial. Particularmente, na propriedade 8., $|\cdot |$ denota a norma (quando existe) definida neste espaço. Esta propriedade é uma extensão natural da desigualdade triangular. No caso do espaço usual dos números reais, $|\cdot |$ é a função valor absoluto.

Para uma sequência $\{x_{i,j}\}_{(i,j)\in \mathbb {N} \times \mathbb {N} }$ é usual denotarmos somatórios duplos da seguinte forma:

\sum _{i,j=1}^{n}:=\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i,j}

Neste contexto temos as seguintes propriedades:

1. $\sum _{k\leq j\leq i\leq n}x_{i,j}:=\sum _{i=k}^{n}\sum _{j=k}^{i}x_{i,j}=\sum _{j=k}^{n}\sum _{i=j}^{n}x_{i,j}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

2. $\sum _{i=m+k}^{n}\sum _{j=m}^{i-k}a_{i,j}=\sum _{j=m}^{n-k}\sum _{i=j+k}^{n}a_{i,j}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

Algumas propriedades envolvendo soma e produto podem ser generalizadas usando a notação de somatório e produtório. Dada uma sequência $\{x_{i}\}_{i\in \mathbb {N} },$ o produtório é, usualmente, denotado por:

\prod _{i=1}^{n}x_{i}:=x_{1}x_{2}x_{3}\cdots x_{n}

Por exemplo, temos as propriedades:

1. $\sum _{n=s}^{t}\ln x_{n}=\ln \prod _{n=s}^{t}x_{n}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

2. $c^{\left[\sum _{n=s}^{t}x_{n}\right]}=\prod _{n=s}^{t}c^{x_{n}},{\mbox{onde}}~c>0.$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

${\left(x+y\right)}^{n}=\sum _{{k=0}}^{n}{n \choose k}x^{{n-k}}y^{k}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$

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