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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 [ editar ]
∑
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}\,\!}
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}}
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)}
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} ^{*}}
B
s
{\displaystyle B_{s}}
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 [ editar ]
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)}
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),\,\!}
s x ) es el polilogaritmo de x .
Denominadores simples [ editar ]
∑
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 [ editar ] 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}}
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}}
∑
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 [ editar ]
∑
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 [ editar ] La serie binomial (incluye la raíz cuadrada para
α
=
1
/
2
{\displaystyle \alpha =1/2}
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 [ editar ]
∑
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 [ editar ] 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 [ editar ]
∑
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}}
Número armónico .
Véase también [ editar ]
↑ Bronshtein, I, y otro (1982). Manual de Matemáticas para ingenieros y estudiantes . Mir. p. 696. ↑ a b c d Theoretical computer science cheat sheet
Referencias [ editar ] 
![{\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}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ace1e8ca0a0e9d694f2181b6c21b377041abbb9f)
