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

  • ∑ i = 1 n i = n ( n + 1 ) 2 = ( n + 1 ) 3 − n 3 − 1 6 {\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}={\frac {(n+1)^{3}-n^{3}-1}{6}}\,\!}
  • ∑ i = p q i = p + ( p + 1 ) + ( p + 2 ) + ( p + 3 ) + … + ( q − 1 ) + q = ( p + q ) ( q − p + 1 ) 2 {\displaystyle \sum _{i=p}^{q}i=p+(p+1)+(p+2)+(p+3)+\ldots +(q-1)+q={\frac {(p+q)(q-p+1)}{2}}\,\!}
  • ∑ i = 1 n 2 i = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + … + ( 2 n − 2 ) + 2 n = n ( n + 1 ) {\displaystyle \sum _{i=1}^{n}2i=2+4+6+8+10+12+14+16+\ldots +(2n-2)+2n=n(n+1)\,\!}
  • ∑ i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 = n 3 3 + n 2 2 + n 6 {\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}}\,\!}
  • ∑ i = 1 n i 3 = ( n ( n + 1 ) 2 ) 2 = n 4 4 + n 3 2 + n 2 4 = [ ∑ i = 1 n i ] 2 {\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}\,\!} (véase: Cuadrados de números triangulares)
  • ∑ i = 1 n i 4 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n − 1 ) 30 {\displaystyle \sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}\,\!}
  • ∑ i = 1 n i 5 = n 2 ( n + 1 ) 2 ( 2 n 2 + 2 n − 1 ) 12 {\displaystyle \sum _{i=1}^{n}i^{5}={\frac {n^{2}(n+1)^{2}(2n^{2}+2n-1)}{12}}\,\!}
  • ∑ i = 1 n i 6 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 4 + 6 n 3 − 3 n + 1 ) 42 {\displaystyle \sum _{i=1}^{n}i^{6}={\frac {n(n+1)(2n+1)(3n^{4}+6n^{3}-3n+1)}{42}}\,\!}
  • ∑ i = 1 n i 7 = n 2 ( n + 1 ) 2 ( 3 n 4 + 6 n 3 − n 2 − 4 n + 2 ) 24 {\displaystyle \sum _{i=1}^{n}i^{7}={\frac {n^{2}(n+1)^{2}(3n^{4}+6n^{3}-n^{2}-4n+2)}{24}}\,\!}
  • ∑ i = 1 n i 8 = n ( n + 1 ) ( 2 n + 1 ) ( 5 n 6 + 15 n 5 + 5 n 4 − 15 n 3 − n 2 + 9 n − 3 ) 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}}\,\!}
  • ∑ i = 1 n i 9 = n 2 ( n + 1 ) 2 ( n 2 + n − 1 ) ( 2 n 4 + 4 n 3 − n 2 − 3 n + 3 ) 20 {\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}}\,\!}
  • ∑ i = 1 n i 10 = n ( n + 1 ) ( 2 n + 1 ) ( n 2 + n − 1 ) ( 3 n 6 + 9 n 5 + 2 n 4 − 11 n 3 + 3 n 2 + 10 n − 5 ) 66 {\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}}\,\!}
  • ∑ i = 1 n i 11 = n 2 ( n + 1 ) 2 ( 2 n 8 + 8 n 7 + 4 n 6 − 16 n 5 − 5 n 4 + 26 n 3 − 3 n 2 − 20 n + 10 ) 24 {\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}}\,\!}
  • ∑ i = 1 n i 12 = n ( n + 1 ) ( 2 n + 1 ) ( 105 n 10 + 525 n 9 + 525 n 8 − 1050 n 7 − 1190 n 6 + 2310 n 5 + 1420 n 4 − 3285 n 3 − 287 n 2 + 2073 n − 691 ) 2730 {\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}}\,\!}
  • ∑ i = 0 n i s = ( n + 1 ) s + 1 s + 1 + ∑ k = 1 s B k s − k + 1 ( s k ) ( n + 1 ) s − k + 1 {\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 B k {\displaystyle B_{k}} es el k-ésimo número de Bernoulli.
  • ∑ i = 1 ∞ i − s = ∏ p  primo 1 1 − p − s = ζ ( s ) {\displaystyle \sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ primo}}}{\frac {1}{1-p^{-s}}}=\zeta (s)\,\!}
donde s > 1 y ζ ( s ) {\displaystyle \zeta (s)} es la función zeta de Riemann.

Series relacionadas con la función zeta de Riemann:

  • ∑ i = 1 ∞ 1 i 2 = π 2 6 , ∑ i = 1 ∞ 1 i 4 = π 4 90 , ∑ i = 1 ∞ 1 i 6 = π 6 945 {\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}}}
  • ∑ i = 1 ∞ 1 i 2 s = ( 2 2 s − 1 ) π 2 s B s ( 2 s ) ! , s ∈ N ∗ {\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 B s {\displaystyle B_{s}} el s-ésimo número de Bernoulli.
  • ∑ i = 1 ∞ ( − 1 ) i + 1 i 2 s = ( 1 − 1 2 2 s − 1 ) ∑ i = 1 ∞ 1 i 2 s {\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}}}}
  • ∑ i = 0 ∞ 1 ( 2 i + 1 ) 2 = π 2 8 , ∑ i = 0 ∞ 1 ( 2 i + 1 ) 4 = π 4 96 , ∑ i = 0 ∞ 1 ( 2 i + 1 ) 6 = π 6 960 {\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[1]​

  • ∑ i = 1 n ( 2 i − 1 ) = 1 + 3 + 5 + 7 + 9 + … + ( 2 n − 3 ) + ( 2 n − 1 ) = n 2 {\displaystyle \sum _{i=1}^{n}(2i-1)=1+3+5+7+9+\ldots +(2n-3)+(2n-1)=n^{2}\,\!}
  • ∑ i = 1 n ( 2 i − 1 ) 2 = 1 2 + 3 2 + 5 2 + 7 2 + 9 2 + … + ( 2 n − 3 ) 2 + ( 2 n − 1 ) 2 = n ( 4 n 2 − 1 ) 3 {\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}}\,\!}
  • ∑ i = 1 n ( 2 i − 1 ) 3 = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + … + ( 2 n − 3 ) 3 + ( 2 n − 1 ) 3 = n 2 ( 2 n 2 − 1 ) {\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 | x | < 1 {\displaystyle |x|<1} ) Suma finita
∑ i = 0 ∞ x i = 1 1 − x {\displaystyle \sum _{i=0}^{\infty }x^{i}={\frac {1}{1-x}}\,\!} ∑ i = 0 n x i = 1 − x n + 1 1 − x = 1 + 1 r ( 1 − 1 ( 1 + r ) n ) {\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 r > 0 {\displaystyle r>0} , x = 1 1 + r . {\displaystyle x={\frac {1}{1+r}}.\,\!}
∑ i = 0 ∞ x 2 i = 1 1 − x 2 {\displaystyle \sum _{i=0}^{\infty }x^{2i}={\frac {1}{1-x^{2}}}\,\!}
∑ i = 1 ∞ i x i = x ( 1 − x ) 2 {\displaystyle \sum _{i=1}^{\infty }ix^{i}={\frac {x}{(1-x)^{2}}}\,\!} ∑ i = 1 n i x i = x 1 − x n ( 1 − x ) 2 − n x n + 1 1 − x {\displaystyle \sum _{i=1}^{n}ix^{i}=x{\frac {1-x^{n}}{(1-x)^{2}}}-{\frac {nx^{n+1}}{1-x}}\,\!}
∑ i = 1 ∞ i 2 x i = x ( 1 + x ) ( 1 − x ) 3 {\displaystyle \sum _{i=1}^{\infty }i^{2}x^{i}={\frac {x(1+x)}{(1-x)^{3}}}\,\!} ∑ i = 1 n i 2 x i = x ( 1 + x − ( n + 1 ) 2 x n + ( 2 n 2 + 2 n − 1 ) x n + 1 − n 2 x n + 2 ) ( 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}}}\,\!}
∑ i = 1 ∞ i 3 x i = x ( 1 + 4 x + x 2 ) ( 1 − x ) 4 {\displaystyle \sum _{i=1}^{\infty }i^{3}x^{i}={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}\,\!}
∑ i = 1 ∞ i 4 x i = x ( 1 + x ) ( 1 + 10 x + x 2 ) ( 1 − x ) 5 {\displaystyle \sum _{i=1}^{\infty }i^{4}x^{i}={\frac {x(1+x)(1+10x+x^{2})}{(1-x)^{5}}}\,\!}
∑ i = 1 ∞ i k x i = Li − k ⁡ ( x ) , {\displaystyle \sum _{i=1}^{\infty }i^{k}x^{i}=\operatorname {Li} _{-k}(x),\,\!} donde Lis(x) es el polilogaritmo de x.

Denominadores simples

  • ∑ n = 1 ∞ x n n = log e ⁡ ( 1 1 − x )  para  | x | ≤ 1 , x ≠ 1 {\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\,\!}
  • ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 x 2 n + 1 = x − x 3 3 + x 5 5 − ⋯ = arctan ⁡ ( x ) {\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)\,\!}
  • ∑ n = 0 ∞ x 2 n + 1 2 n + 1 = a r c t a n h ( x )  para  | x | < 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}=\mathrm {arctanh} (x)\quad {\mbox{ para }}|x|<1\,\!}
  • ∑ n = 1 ∞ 1 n 2 = π 2 6 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}\,\!}
  • ∑ n = 1 ∞ 1 n 4 = π 4 90 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {\pi ^{4}}{90}}\,\!}
  • ∑ n = 1 ∞ y n 2 + y 2 = − 1 2 y + π 2 coth ⁡ ( π y ) {\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.

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


  • ∑ i = 0 ∞ ( − 1 ) i ( 2 i + 1 ) ! x 2 i + 1 = x − x 3 3 ! + x 5 5 ! − ⋯ = sin ⁡ 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}
  • ∑ i = 0 ∞ ( − 1 ) i ( 2 i ) ! x 2 i = 1 − x 2 2 ! + x 4 4 ! − ⋯ = cos ⁡ x {\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}x^{2i}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots =\cos x}
  • ∑ i = 0 ∞ x 2 i + 1 ( 2 i + 1 ) ! = sinh ⁡ x {\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x}
  • ∑ i = 0 ∞ x 2 i ( 2 i ) ! = cosh ⁡ x {\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x}

Denominadores factoriales modificados

  • ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = arcsin ⁡ x  para  | x | < 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=\arcsin x\quad {\mbox{ para }}|x|<1\!}
  • ∑ i = 0 ∞ ( − 1 ) i ( 2 i ) ! 4 i ( i ! ) 2 ( 2 i + 1 ) x 2 i + 1 = a r c s i n h ( x )  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 α = 1 / 2 {\displaystyle \alpha =1/2} y la serie geométrica infinita para α = − 1 {\displaystyle \alpha =-1} ):

raíz cuadrada:

  • 1 + x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! ( 1 − 2 n ) n ! 2 4 n x n  para  | x | < 1 {\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:

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

Forma general:

  • ( 1 + x ) α = ∑ n = 0 ∞ ( α n ) x n  para todo  | x | < 1  y todo complejo  α {\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
( α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯ ( α − n + 1 ) n ! {\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!}
  • [2]​ ∑ i = 0 ∞ ( i + n i ) x i = 1 ( 1 − x ) n + 1 {\displaystyle \sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}}
  • [2]​ ∑ i = 0 ∞ 1 i + 1 ( 2 i i ) x i = 1 2 x ( 1 − 4 x ) {\displaystyle \sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}({\sqrt {1-4x}})}
  • [2]​ ∑ i = 0 ∞ ( 2 i i ) x i = 1 1 − 4 x {\displaystyle \sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}}
  • [2]​ ∑ i = 0 ∞ ( 2 i + n i ) x i = 1 1 − 4 x ( 1 − 1 − 4 x 2 x ) n {\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

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

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

Sin clasificar

  • ∑ n = b + 1 ∞ b n 2 − b 2 = 1 2 H 2 b {\displaystyle \sum _{n=b+1}^{\infty }{\frac {b}{n^{2}-b^{2}}}={\frac {1}{2}}H_{2b}}

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

Véase también

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