symsum

Symbolic sum of series

Syntax

F = symsum(f,k,a,b)

F = symsum(f,k)

Description

F = symsum(f,k,a,b) returns the symbolic definite sum of the series f with respect to the summation index k from the lower bound a to the upper bound b. If you do not specify k, symsum uses the variable determined by symvar as the summation index. If f is a constant, then the default variable is x.

symsum(f,k,[a b]) or symsum(f,k,[a; b]) is equivalent to symsum(f,k,a,b).

F = symsum(f,k) returns the indefinite sum (antidifference) of the series f with respect to the summation index k. The f argument defines the series such that the indefinite sum F satisfies the relation F(k+1) - F(k) = f(k). If you do not specify k, symsum uses the variable determined by symvar as the summation index. If f is a constant, then the default variable is x.

Examples

Find Sum of Sequence of Numbers

Find the sum of the integer numbers $1+2+3+...+\mathit{n}={\sum }_{\mathit{k}=1}^{\mathit{n}}\mathit{k}$.

syms k n
F1 = symsum(k,k,1,n)

F1 = 
$$\frac{n\hspace{0.17em}\left(n+1\right)}{2}$$

Find the sum of the square numbers $1^2+2^2+3^2+...+{\mathit{n}}^2={\sum }_{\mathit{k}=1}^{\mathit{n}}{\mathit{k}}^2$.

syms k n
F2 = symsum(k^2,k,1,n)

F2 = 
$$\frac{n\hspace{0.17em}\left(2\hspace{0.17em}n+1\right)\hspace{0.17em}\left(n+1\right)}{6}$$

Find the sum of the cubic numbers $1^3+2^3+3^3+...+{\mathit{n}}^3={\sum }_{\mathit{k}=1}^{\mathit{n}}{\mathit{k}}^3$.

syms k n
F3 = symsum(k^3,k,1,n)

F3 = 
$$\frac{n^2\hspace{0.17em}{\left(n+1\right)}^2}{4}$$

Find Sum of Series with Bounds

Find the following sums of series.

$$\begin{array}{l}\mathrm{F1}=\sum _{\mathit{k}=0}^{10}{\mathit{k}}^2\\ \mathrm{F2}=\sum _{\mathit{k}=1}^{\infty }\frac{1}{{\mathit{k}}^2}\\ \mathrm{F3}=\sum _{\mathit{k}=1}^{\infty }\frac{{\mathit{x}}^{\mathit{k}}}{\mathit{k}!}\end{array}$$

syms k x
F1 = symsum(k^2,k,0,10)

F1 = $$385$$

F2 = symsum(1/k^2,k,1,Inf)

F2 = 
$$\frac{{\pi }^2}{6}$$

F3 = symsum(x^k/factorial(k),k,1,Inf)

F3 = $${\mathrm{e}}^x-1$$

Alternatively, you can specify summation bounds as a row or column vector.

F1 = symsum(k^2,k,[0 10])

F1 = $$385$$

F2 = symsum(1/k^2,k,[1;Inf])

F2 = 
$$\frac{{\pi }^2}{6}$$

F3 = symsum(x^k/factorial(k),k,[1 Inf])

F3 = $${\mathrm{e}}^x-1$$

Find Indefinite Sum of Series

Find the following indefinite sums of series (antidifferences).

$$\begin{array}{l}\mathrm{F1}=\sum _{\mathit{k}}\mathit{k}\\ \mathrm{F2}=\sum _{\mathit{k}}{2}^{\mathit{k}}\\ \mathrm{F3}=\sum _{\mathit{k}}\frac{1}{{\mathit{k}}^2}\end{array}$$

syms k
F1 = symsum(k,k)

F1 = 
$$\frac{k^2}{2}-\frac{k}{2}$$

F2 = symsum(2^k,k)

F2 = $$2^k$$

F3 = symsum(1/k^2,k)

F3 = 
$$\{\begin{array}{cl}-{\psi \text{<a href="matlab:doc symbolic/psi">psi</a>}}^{\prime }\left(k\right)& \text{ if }0<k\\ {\psi \text{<a href="matlab:doc symbolic/psi">psi</a>}}^{\prime }\left(1-k\right)& \text{ if }k\le 0\end{array}$$

Summation of Polynomial Series

Find the summation of the polynomial series $\mathit{F}\left(\mathit{x}\right)={\sum }_{\mathit{k}=1}^8{\mathit{a}}_{\mathit{k}}{\mathit{x}}^{\mathit{k}}$.

If you know that the coefficient $$a_k$$ is a function of some integer variable $$k$$, use the symsum function. For example, find the sum $\mathit{F}\left(\mathit{x}\right)={\sum }_{\mathit{k}=1}^8\mathit{k}{\mathit{x}}^{\mathit{k}}$.

syms x k
F(x) = symsum(k*x^k,k,1,8)

F(x) = $$8\hspace{0.17em}{x}^8+7\hspace{0.17em}{x}^7+6\hspace{0.17em}{x}^6+5\hspace{0.17em}{x}^5+4\hspace{0.17em}{x}^4+3\hspace{0.17em}{x}^3+2\hspace{0.17em}{x}^2+x$$

Calculate the summation series for $$x=2$$.

F(2)

ans = $$3586$$

Alternatively, if you know that the coefficients $$a_k$$ are a vector of values, you can use the sum function. For example, the coefficients are ${\mathit{a}}_1,\dots ,{\mathit{a}}_8=1,\dots ,8$. Declare the term $$x^k$$ as a vector by using subs(x^k,k,1:8).

a = 1:8;
G(x) = sum(a.*subs(x^k,k,1:8))

G(x) = $$8\hspace{0.17em}{x}^8+7\hspace{0.17em}{x}^7+6\hspace{0.17em}{x}^6+5\hspace{0.17em}{x}^5+4\hspace{0.17em}{x}^4+3\hspace{0.17em}{x}^3+2\hspace{0.17em}{x}^2+x$$

Calculate the summation series for $$x=2$$.

G(2)

ans = $$3586$$

Input Arguments

f — Expression defining terms of series

symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic number

Expression defining terms of series, specified as a symbolic expression, function, vector, matrix, or symbolic number.

k — Summation index

symbolic variable

Summation index, specified as a symbolic variable. If you do not specify this variable, symsum uses the default variable determined by symvar(expr,1). If f is a constant, then the default variable is x.

a — Lower bound of summation index

number | symbolic number | symbolic variable | symbolic expression | symbolic function

Lower bound of the summation index, specified as a number, symbolic number, variable, expression, or function (including expressions and functions with infinities).

b — Upper bound of summation index

number | symbolic number | symbolic variable | symbolic expression | symbolic function

Upper bound of the summation index, specified as a number, symbolic number, variable, expression, or function (including expressions and functions with infinities).

More About

Definite Sum

The definite sum of a series is defined as

$${\displaystyle \sum _{k=a}^b{f}_k}=f_a+f_{a+1}+\dots +f_b.$$

Indefinite Sum

The indefinite sum (antidifference) of a series is defined as

$$F(x)={\displaystyle \sum _{x}f\left(x\right)},$$

such that

$$F\left(x+1\right)-F\left(x\right)=f\left(x\right).$$

Version History

Introduced before R2006a