Anexo:Series matemáticas

Esta lista de series matemáticas contiene fórmulas para sumatorias finitas e infinitas. Puede ser usada junto con otras herramientas para evaluar sumas.

Sumatoria de potencias

• ${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}={\frac {(n+1)^{3}-n^{3}-1}{6}}\,\!}$
• ${\displaystyle \sum _{i=p}^{q}i=p+(p+1)+(p+2)+(p+3)+\ldots +(q-1)+q={\frac {(p+q)(q-p+1)}{2}}\,\!}$

• ${\displaystyle \sum _{i=1}^{n}2i=2+4+6+8+10+12+14+16+\ldots +(2n-2)+2n=n(n+1)\,\!}$

• ${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{3}=\left({\frac {n(n+1)}{2}}\right)^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}=\left^{2}\,\!}$ (véase: Cuadrados de números triangulares)
• ${\displaystyle \sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{5}={\frac {n^{2}(n+1)^{2}(2n^{2}+2n-1)}{12}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{6}={\frac {n(n+1)(2n+1)(3n^{4}+6n^{3}-3n+1)}{42}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{7}={\frac {n^{2}(n+1)^{2}(3n^{4}+6n^{3}-n^{2}-4n+2)}{24}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{8}={\frac {n(n+1)(2n+1)(5n^{6}+15n^{5}+5n^{4}-15n^{3}-n^{2}+9n-3)}{24}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{9}={\frac {n^{2}(n+1)^{2}(n^{2}+n-1)(2n^{4}+4n^{3}-n^{2}-3n+3)}{20}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{10}={\frac {n(n+1)(2n+1)(n^{2}+n-1)(3n^{6}+9n^{5}+2n^{4}-11n^{3}+3n^{2}+10n-5)}{66}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{11}={\frac {n^{2}(n+1)^{2}(2n^{8}+8n^{7}+4n^{6}-16n^{5}-5n^{4}+26n^{3}-3n^{2}-20n+10)}{24}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}i^{12}={\frac {n(n+1)(2n+1)(105n^{10}+525n^{9}+525n^{8}-1050n^{7}-1190n^{6}+2310n^{5}+1420n^{4}-3285n^{3}-287n^{2}+2073n-691)}{2730}}\,\!}$
• ${\displaystyle \sum _{i=0}^{n}i^{s}={\frac {(n+1)^{s+1}}{s+1}}+\sum _{k=1}^{s}{\frac {B_{k}}{s-k+1}}{s \choose k}(n+1)^{s-k+1}\,\!}$

donde ${\displaystyle B_{k}}$ es el k-ésimo número de Bernoulli.

• ${\displaystyle \sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ primo}}}{\frac {1}{1-p^{-s}}}=\zeta (s)\,\!}$

donde s > 1 y ${\displaystyle \zeta (s)}$ es la función zeta de Riemann.

Series relacionadas con la función zeta de Riemann:

• ${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i^{2}}}={\frac {\pi ^{2}}{6}},\qquad \sum _{i=1}^{\infty }{\frac {1}{i^{4}}}={\frac {\pi ^{4}}{90}},\qquad \sum _{i=1}^{\infty }{\frac {1}{i^{6}}}={\frac {\pi ^{6}}{945}}}$
• ${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i^{2s}}}={\frac {(2^{2s-1})\pi ^{2s}B_{s}}{(2s)!}},\quad s\in \mathbb {N} ^{*}}$ y siendo ${\displaystyle B_{s}}$ el s-ésimo número de Bernoulli.
• ${\displaystyle \sum _{i=1}^{\infty }{\frac {(-1)^{i+1}}{i^{2s}}}=\left(1-{\frac {1}{2^{2s-1}}}\right)\sum _{i=1}^{\infty }{\frac {1}{i^{2s}}}}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {1}{(2i+1)^{2}}}={\frac {\pi ^{2}}{8}},\qquad \sum _{i=0}^{\infty }{\frac {1}{(2i+1)^{4}}}={\frac {\pi ^{4}}{96}},\qquad \sum _{i=0}^{\infty }{\frac {1}{(2i+1)^{6}}}={\frac {\pi ^{6}}{960}}}$

Otras sumas numéricas son​

• ${\displaystyle \sum _{i=1}^{n}(2i-1)=1+3+5+7+9+\ldots +(2n-3)+(2n-1)=n^{2}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}(2i-1)^{2}=1^{2}+3^{2}+5^{2}+7^{2}+9^{2}+\ldots +(2n-3)^{2}+(2n-1)^{2}={\frac {n(4n^{2}-1)}{3}}\,\!}$
• ${\displaystyle \sum _{i=1}^{n}(2i-1)^{3}=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+\ldots +(2n-3)^{3}+(2n-1)^{3}=n^{2}(2n^{2}-1)\,\!}$

Serie de potencias

Suma infinita (para ${\displaystyle |x|<1}$)	Suma finita
${\displaystyle \sum _{i=0}^{\infty }x^{i}={\frac {1}{1-x}}\,\!}$	${\displaystyle \sum _{i=0}^{n}x^{i}={\frac {1-x^{n+1}}{1-x}}=1+{\frac {1}{r}}\left(1-{\frac {1}{(1+r)^{n}}}\right)}$ donde ${\displaystyle r>0}$ , ${\displaystyle x={\frac {1}{1+r}}.\,\!}$
${\displaystyle \sum _{i=0}^{\infty }x^{2i}={\frac {1}{1-x^{2}}}\,\!}$
${\displaystyle \sum _{i=1}^{\infty }ix^{i}={\frac {x}{(1-x)^{2}}}\,\!}$	${\displaystyle \sum _{i=1}^{n}ix^{i}=x{\frac {1-x^{n}}{(1-x)^{2}}}-{\frac {nx^{n+1}}{1-x}}\,\!}$
${\displaystyle \sum _{i=1}^{\infty }i^{2}x^{i}={\frac {x(1+x)}{(1-x)^{3}}}\,\!}$	${\displaystyle \sum _{i=1}^{n}i^{2}x^{i}={\frac {x(1+x-(n+1)^{2}x^{n}+(2n^{2}+2n-1)x^{n+1}-n^{2}x^{n+2})}{(1-x)^{3}}}\,\!}$
${\displaystyle \sum _{i=1}^{\infty }i^{3}x^{i}={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}\,\!}$
${\displaystyle \sum _{i=1}^{\infty }i^{4}x^{i}={\frac {x(1+x)(1+10x+x^{2})}{(1-x)^{5}}}\,\!}$
${\displaystyle \sum _{i=1}^{\infty }i^{k}x^{i}=\operatorname {Li} _{-k}(x),\,\!}$ donde Lis(x) es el polilogaritmo de x.

Denominadores simples

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=\log _{e}\left({\frac {1}{1-x}}\right)\quad {\mbox{ para }}|x|\leq 1,\,x\not =1\,\!}$

• ${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots =\arctan(x)\,\!}$

• ${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}=\mathrm {arctanh} (x)\quad {\mbox{ para }}|x|<1\,\!}$

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}\,\!}$
• ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {\pi ^{4}}{90}}\,\!}$

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {y}{n^{2}+y^{2}}}=-{\frac {1}{2y}}+{\frac {\pi }{2}}\coth(\pi y)}$

Denominadores factoriales

Muchas series de potencias originadas del Teorema de Taylor tienen un coeficiente conteniendo un factorial.

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}$

• ${\displaystyle \sum _{i=0}^{\infty }i{\frac {x^{i}}{i!}}=xe^{x}}$ (c.f. media de la distribución de Poisson)
• ${\displaystyle \sum _{i=0}^{\infty }i^{2}{\frac {x^{i}}{i!}}=(x+x^{2})e^{x}}$ (c.f. segundo momento de la distribución de Poisson)
• ${\displaystyle \sum _{i=0}^{\infty }i^{3}{\frac {x^{i}}{i!}}=(x+3x^{2}+x^{3})e^{x}}$
• ${\displaystyle \sum _{i=0}^{\infty }i^{4}{\frac {x^{i}}{i!}}=(x+7x^{2}+6x^{3}+x^{4})e^{x}}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}x^{2i+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots =\sin x}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}x^{2i}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots =\cos x}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x}$

Denominadores factoriales modificados

• ${\displaystyle \sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=\arcsin x\quad {\mbox{ para }}|x|<1\!}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{4^{i}(i!)^{2}(2i+1)}}x^{2i+1}=\mathrm {arcsinh} (x)\quad {\mbox{ para }}|x|<1\!}$

Serie binomial

La serie binomial (incluye la raíz cuadrada para ${\displaystyle \alpha =1/2}$ y la serie geométrica infinita para ${\displaystyle \alpha =-1}$):

raíz cuadrada:

• ${\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)n!^{2}4^{n}}}x^{n}\quad {\mbox{ para }}|x|<1\!}$

serie geométrica:

• ${\displaystyle (1+x)^{-1}=\sum _{n=0}^{\infty }(-1)^{n}x^{n}\quad {\mbox{ para }}|x|<1}$

Forma general:

• ${\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\alpha \choose n}x^{n}\quad {\mbox{ para todo }}|x|<1{\mbox{ y todo complejo }}\alpha \!}$

con coeficientes binomiales generalizados

${\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!}$

• ​ ${\displaystyle \sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}}$
• ​ ${\displaystyle \sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}({\sqrt {1-4x}})}$
• ​ ${\displaystyle \sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}}$
• ​ ${\displaystyle \sum _{i=0}^{\infty }{2i+n \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}\left({\frac {1-{\sqrt {1-4x}}}{2x}}\right)^{n}}$

Coeficientes binomiales

• ${\displaystyle \sum _{i=0}^{n}{n \choose i}=2^{n}}$
• ${\displaystyle \sum _{i=0}^{n}{n \choose i}a^{(n-i)}b^{i}=(a+b)^{n}}$
• ${\displaystyle \sum _{i=0}^{2n+1}(-1)^{i}{2n+1 \choose i}=0}$
• ${\displaystyle \sum _{i=0}^{n}{i \choose k}={n+1 \choose k+1}}$
• ${\displaystyle \sum _{i=0}^{n}{k+i \choose i}={k+n+1 \choose n}}$
• ${\displaystyle \sum _{i=0}^{r}{r \choose i}{s \choose n-i}={r+s \choose n}}$

Funciones trigonométricas

La sumatoria de senos y cosenos se originan en la serie de Fourier.

• ${\displaystyle \sum _{i=1}^{n}\sin \left({\frac {i\pi }{n}}\right)=0}$
• ${\displaystyle \sum _{i=1}^{n}\cos \left({\frac {i\pi }{n}}\right)=0}$

Sin clasificar

• ${\displaystyle \sum _{n=b+1}^{\infty }{\frac {b}{n^{2}-b^{2}}}={\frac {1}{2}}H_{2b}}$

Para el significado de ${\displaystyle H_{k}}$ véase Número armónico.

Véase también

• Serie matemática
• Anexo:Lista de integrales
• Serie de Taylor
• Teorema del binomio
• 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + …

Notas

1. ↑ Bronshtein, I, y otro (1982). Manual de Matemáticas para ingenieros y estudiantes. Mir. p. 696. 
2. ↑ ^{a b c d} Theoretical computer science cheat sheet

Referencias

• Varios libros con lista de integrales tienen también una lista de series.