symprod

Product of series

Syntax

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

F = symprod(f,k)

Description

F = symprod(f,k,a,b) returns the product of the series with terms that expression f specifies, which depend on symbolic variable k. The value of k ranges from a to b. If you do not specify k, symprod uses the variable that symvar determines. If f is a constant, then the default variable is x.

F = symprod(f,k) returns the product of the series that expression f specifies, which depend on symbolic variable k. The value of k starts at 1 with an unspecified upper bound. The product F is returned in terms of k where k represents the upper bound. This product F differs from the indefinite product. If you do not specify k, symprod uses the variable that symvar determines. If f is a constant, then the default variable is x.

Examples

Find Product of Series Specifying Bounds

Find the following products of series

$$\begin{array}{l}P1={\displaystyle \prod _{k=2}^{\infty }1-\frac{1}{k^2}},\\ P2={\displaystyle \prod _{k=2}^{\infty }\frac{k^2}{k^2-1}}.\end{array}$$

syms k
P1 = symprod(1 - 1/k^2, k, 2, Inf)
P2 = symprod(k^2/(k^2 - 1), k, 2, Inf)

P1 =
1/2
P2 =
2

Alternatively, specify bounds as a row or column vector.

syms k
P1 = symprod(1 - 1/k^2, k, [2 Inf])
P2 = symprod(k^2/(k^2 - 1), k, [2; Inf])

P1 =
1/2
P2 =
2

Find Product of Series Specifying Product Index and Bounds

Find the product of series

$$P={\displaystyle \prod _{k=1}^{10000}\frac{e^{kx}}{x}}.$$

syms k x
P = symprod(exp(k*x)/x, k, 1, 10000)

P =
exp(50005000*x)/x^10000

Find Product of Series with Unspecified Bounds

When you do not specify the bounds of a series are unspecified, the variable k starts at 1. In the returned expression, k itself represents the upper bound.

Find the products of series with an unspecified upper bound

$$\begin{array}{l}P1={\displaystyle \prod _{k}k},\\ P2={\displaystyle \prod _k\frac{2k-1}{k^2}}.\end{array}$$

syms k
P1 = symprod(k, k)
P2 = symprod((2*k - 1)/k^2, k)

P1 =
factorial(k)
P2 =
(1/2^(2*k)*2^(k + 1)*factorial(2*k))/(2*factorial(k)^3)

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, constant, or a vector or matrix of symbolic expressions, functions, or constants.

k — Product index

symbolic variable

Product index, specified as a symbolic variable. If you do not specify this variable, symprod uses the default variable that symvar(expr,1) determines. If f is a constant, then the default variable is x.

a — Lower bound of product index

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

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

b — Upper bound of product index

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

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

More About

Definite Product

The definite product of a series is defined as

$${\displaystyle \prod _{i=a}^b{x}_i}=x_a\cdot x_{a+1}\cdot \dots \cdot x_b$$

Indefinite Product

The indefinite product of x_i over i is

$$f(i)={\displaystyle \prod _i{x}_i}$$

This definition holds under the assumption that the following identity is true for all values of i.

$$\frac{f\left(i+1\right)}{f\left(i\right)}=x_i$$

Note

symprod does not compute indefinite products.