# symprod

Product of series

## Syntax

• `symprod(expr,var)` example
• `symprod(expr,var,a,b)` example

## Description

example

````symprod(expr,var)` evaluates the product of a series, where expression `expr` defines the terms of a series, with respect to the symbolic variable `var`. The value of the variable `var` changes from 1 to `var`. If you do not specify the variable, `symprod` uses the default variable determined by `symvar`. If `expr` is a constant, then the default variable is `x`.```

example

````symprod(expr,var,a,b)` evaluates the product of a series, where expression `expr` defines the terms of a series, with respect to the symbolic variable `var`. The value of the variable `var` changes from `a` to `b`. If you do not specify the variable, `symprod` uses the default variable determined by `symvar`. If `expr` is a constant, then the default variable is `x`.`symprod(expr,var,[a,b])`, ```symprod(expr,var,[a b])```, and `symprod(expr,var,[a;b])` are equivalent to `symprod(expr,var,a,b)`.```

## Examples

### Evaluate Product of Series

Evaluate the product of a series for the symbolic expressions `k` and `k^2`:

```syms k symprod(k) symprod((2*k - 1)/k^2)```
```ans = factorial(k) ans = (1/2^(2*k)*2^(k + 1)*factorial(2*k))/(2*factorial(k)^3)```

### Evaluate Product of Series Specifying Bounds

Evaluate the product of a series for these expressions specifying the bounds:

```syms k symprod(1 - 1/k^2, k, 2, Inf) symprod(k^2/(k^2 - 1), k, 2, Inf)```
```ans = 1/2 ans = 2```

You also can specify bounds as a row or column vector:

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

### Evaluate Product of Series Specifying Product Index and Bounds

Evaluate the product of a series for this multivariable expression with respect to `k`:

```syms k x symprod(exp(k*x)/x, k, 1, 10000)```
```ans = exp(50005000*x)/x^10000```

## Input Arguments

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### `expr` — Expression defining terms of seriessymbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic number

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

### `var` — Product indexsymbolic variable

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

### `a` — Lower bound of product indexnumber | 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 indexnumber | 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).

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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.