Definición

Una sumatoria es una forma abrevidada de escribir la suma de números; esta suma abreviada tiene una simbología propia, haciendo uso de un símbolo \(\Sigma\), unos límites: inferior \(i=1\) inicializado por medio del contador \(i\), y un límite superior \(n\geq1\) un número entero; y finalmente un sumando o sumandos, valores a ser sumados \(x_i\in\mathbf{R}\) con \(i=1,2,\ldots,n\)

\[\sum_{i=1}^{n}x_i=x_1+x_2+\cdots+x_n\]

Nota

Los sumandos pueden tomar cualquier valor, aunque se puede definir una regla que permita establecer que numeros quieren sumarse.

Ejemplo

Sean \(x_1=5\), \(x_2=3\), \(x_3=7\), \(x_4=1\) y \(x_5=-6\) y lo que se pretende es sumar estos valores

\[\sum_{i=1}^{5}x_i=5+3+7+1+(-6)\]

5+3+7+1+(-6)
## [1] 10
x <- c(5, 3, 7, 1, -6)
sum(x)
## [1] 10
x[1]+x[2]+x[3]+x[4]+x[5]
## [1] 10

Propiedades

Dentro de la sumatoria se cumplen algunas propiedades, útiles a la hora de realizar sumas o entender como funcionan en sí mismas. Se destacaran solo algunas:

Propiedad

\[\sum_{i=1}^{n}k=nk.\notag\ n\in\textbf{Z}^{+}, k\ una\ constante\]

Ejemplo

\[\sum_{i=1}^{5}4=5\cdot4=20\]

4+4+4+4+4
## [1] 20
y <- c(4, 4, 4, 4, 4)
sum(y)
## [1] 20

Ejemplo

\[\sum_{i=1}^{518}4=518\cdot4=2072\]

z <- rep(x=4, times=518)
z
##   [1] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##  [38] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##  [75] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [112] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [149] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [186] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [223] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [260] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [297] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [334] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [371] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [408] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [445] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [482] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
sum(z)
## [1] 2072
sum(rep(x=4,times=518))
## [1] 2072

Propiedad

\[\sum_{i=1}^{n}i=\frac{n\left(n+1\right)}{2}.\notag\ n\in\textbf{Z}^{+}\]

Ejemplo

\[\sum_{i=1}^{100}i=\frac{100\left(100+1\right)}{2}=5050\]

\[\begin{array}{ccccccccccc} 100 & 99 & 98 & 97 & \cdots & 2 & 1 & = & 100*101/2 & = & 5050\\ 1 & 2 & 3 & 4 & \cdots & 99 & 100 & = & 100*101/2 & = & 5050\\ --- & --- & --- & --- & \cdots & --- & --- & & --------- & & -----\\ 101 & 101 & 101 & 101 & \cdots & 101 & 101 & = & 100*101 & = & 10100 \end{array}\] 
w <- 1:100
sum(w)
## [1] 5050

Ejemplo

\[\sum_{i=1}^{10}i=\frac{10\left(10+1\right)}{2}=55\]

\[\begin{array}{cccccccccccccc} 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & = & 10*11/2 & = & 55\\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & = & 10*11/2 & = & 55\\ -- & -- & -- & -- & -- & -- & -- & -- & -- & -- & & ------- & & ---\\ 11 & 11 & 11 & 11 & 11 & 11 & 11 & 11 & 11 & 11 & = & 10*11 & = & 110 \end{array}\] 
v <- 1:10
sum(v)
## [1] 55

Ejemplo

\[\sum_{i=1}^{50}i=\frac{50\left(50+1\right)}{2}=1275\]

sum(1:50)
## [1] 1275

Propiedad

\[\sum_{i=1}^{n}i^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}.\notag\ n\in\textbf{Z}^{+}\]

Ejemplo

\[\sum_{i=1}^{4}i^{2}=\frac{4\left(4+1\right)\left(2\cdot4+1\right)}{6}=\frac{4\cdot5\cdot(8+1)}{6}=\frac{20\cdot9}{6}=\frac{180}{6}=30\]

h <- (1:4)^2
sum(h)
## [1] 30

Ejemplo

\[\sum_{i=1}^{9}i^{2}=\frac{9\left(9+1\right)\left(2\cdot9+1\right)}{6}=\frac{9\cdot10\cdot(18+1)}{6}=\frac{90\cdot19}{6}=\frac{1710}{6}=285\]

i <- (1:9)^2
sum(i)
## [1] 285

Ejemplo

\[\sum_{i=1}^{5}i^{2}=\frac{5\left(5+1\right)\left(2\cdot5+1\right)}{6}=\frac{5\cdot6(10+1)}{6}=\frac{30\cdot11}{6}=\frac{330}{6}=55\]

a <- (1:5)**2
sum(a)
## [1] 55

Ejemplo

\[\sum_{i=1}^{10}i^{2}=\frac{10\left(10+1\right)\left(2\cdot10+1\right)}{6}=\frac{10\cdot11(20+1)}{6}=\frac{110\cdot21}{6}=\frac{2310}{6}=385\]

b <- (1:10)**2
sum(b)
## [1] 385

Ejemplo

\[\sum_{i=1}^{200}i^{2}=\frac{200\left(200+1\right)\left(2\cdot200+1\right)}{6}=\frac{200\cdot201(400+1)}{6}=\frac{40200\cdot401}{6}=\frac{16120200}{6}=2686700\]

j <- (1:200)^2
sum(j)
## [1] 2686700

Propiedad

\[\sum_{i=1}^{n}i^{3}=\left[\frac{n\left(n+1\right)}{2}\right]^{2}.\notag\ n\in\textbf{Z}^{+}\]

Ejemplo

\[\sum_{i=1}^{5}i^{3}=\left[\frac{5\left(5+1\right)}{2}\right]^{2}=\left(\frac{5\cdot6}{2}\right)^2=\left(\frac{30}{2}\right)^2=15^2=225\]

k <- (1:5)^3
sum(k)
## [1] 225

Ejemplo

\[\sum_{i=1}^{7}i^{3}=\left[\frac{7\left(7+1\right)}{2}\right]^{2}=\left(\frac{7\cdot8}{2}\right)^{2}=\left(\frac{56}{2}\right)^{2}=28^2=784\]

c <- (1:7)**3
sum(c)
## [1] 784

Ejemplo

\[\sum_{i=1}^{10}i^{3}=\left[\frac{10\left(10+1\right)}{2}\right]^{2}=\left(\frac{10\cdot11}{2}\right)^{2}=\left(\frac{110}{2}\right)^{2}=55^2=3025\]

d <- (1:10)**3
sum(d)
## [1] 3025

Propiedad

\[\sum_{i=1}^{n}i^{4}=\frac{n\left(n+1\right)\left(2n+1\right)\left(3n^{2}+3n-1\right)}{30}.\notag\ n\in\textbf{Z}^{+}\]

Ejemplo

\[\sum_{i=1}^{4}i^{4}=\frac{4\left(4+1\right)\left(2\cdot4+1\right)\left(3\cdot4^{2}+3\cdot4-1\right)}{30}=\frac{4\cdot5(8+1)(3\cdot16+12-1)}{30}=\frac{20\cdot9(48+11)}{30}=\frac{180\cdot59}{30}=\frac{10620}{30}=354\]

e <- (1:4)**4
sum(e)
## [1] 354

Ejemplo

\[\sum_{i=1}^{9}i^{4}=\frac{9\left(9+1\right)\left(2\cdot9+1\right)\left(3\cdot9^{2}+3\cdot9-1\right)}{30}=\frac{9\cdot10(18+1)(3\cdot81+27-1)}{30}=\frac{90\cdot19(243+26)}{30}=\frac{1710\cdot269}{30}=\frac{459990}{30}=15333\]

f <- (1:9)**4
sum(f)
## [1] 15333

Propiedad

\[\sum_{i=1}^{n}ka_{i}=k\sum_{i=1}^{n}a_{i}.\notag\ n\in\textbf{Z}^{+}, k\ una\ constante\]

Ejemplo

\[\sum_{i=1}^{9}10{\cdot}i^{4}=10\sum_{i=1}^{9}i^{4}=10\cdot15333=153330\]

sum(10*f)
## [1] 153330
10*sum(f)
## [1] 153330

Ejemplo

\[\sum_{i=1}^{10}3{\cdot}i^{3}=3\sum_{i=1}^{10}i^{3}=3\cdot3025=9075\]

sum(3*d)
## [1] 9075
3*sum(d)
## [1] 9075

Propiedad

\[\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\sum_{i=1}^{n}x_{i}+\sum_{i=1}^{n}y_{i}.\notag\ x_{i},y_{i}\in\textbf{R}\ y\ n\in\textbf{Z}^{+}\]

Ejemplo

\[\sum_{i=1}^{5}\left(i^{2}+i^{3}\right)=\sum_{i=1}^{5}i^{2}+\sum_{i=1}^{5}i^{3}=\frac{5(5+1)(2\cdot5+1)}{6}+\left[\frac{5(5+1)}{2}\right]^{2}=\frac{5\cdot6(10+1)}{6}+\left(\frac{5\cdot6}{2}\right)^{2}=\frac{30\cdot11}{6}+\left(\frac{30}{2}\right)^{2}=\frac{330}{6}+15^2=55+225=280\]

g <- 1:5
h <- g**2
i <- g**3
sum((1:5)**2+(1:5)**3)
## [1] 280
sum(h+i)
## [1] 280
sum(h)+sum(i)
## [1] 280

Ejemplo

\[\sum_{i=1}^{7}\left(i+i^{3}\right)=\sum_{i=1}^{7}i+\sum_{i=1}^{7}i^{3}=\frac{7(7+1)}{2}+\left[\frac{7(7+1)}{2}\right]^{2}=\frac{7\cdot8}{2}+\left(\frac{7\cdot8}{2}\right)^{2}=\frac{56}{2}+\left(\frac{56}{2}\right)^{2}=28+28^2=28+784=812\]

l <- 1:7
m <- l**3
sum(l+m)
## [1] 812
sum(l) + sum(m)
## [1] 812

Ejemplo

\[\sum_{i=1}^{10}\left(i+i^{2}\right)=\sum_{i=1}^{10}i+\sum_{i=1}^{10}i^{2}=\frac{10(10+1)}{2}+\frac{10(10+1)(2\cdot10+1)}{6}=\frac{10\cdot11}{2}+\frac{10\cdot11(20+1)}{6}=\frac{110}{2}+\frac{110\cdot21}{6}=55+385=440\]

n <- 1:10
o <- n
p <- n**2
sum(o+p)
## [1] 440
sum(o) + sum(p)
## [1] 440