# Documentation

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# symprod

Product of series

## Syntax

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

## Description

example

````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`.```

example

````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

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

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

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

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Expression defining terms of series, specified as a symbolic expression, function, constant, or a vector or matrix of symbolic expressions, functions, or constants.

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`.

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

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

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### Definite Product

The definite product of a series is defined as

`$\prod _{i=a}^{b}{x}_{i}={x}_{a}\cdot {x}_{a+1}\cdot \dots \cdot {x}_{b}$`

### Indefinite Product

`$f\left(i\right)=\prod _{i}{x}_{i}$`

is called the indefinite product of xi over i, if the following identity holds for all values of i:

`$\frac{f\left(i+1\right)}{f\left(i\right)}={x}_{i}$`
 Note:   `symprod` does not compute indefinite products.