# symsum

## Syntax

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

## Description

example

````F = symsum(f,k,a,b)` returns the 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)`.```

example

````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

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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 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 =  ```

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) = $8 {x}^{8}+7 {x}^{7}+6 {x}^{6}+5 {x}^{5}+4 {x}^{4}+3 {x}^{3}+2 {x}^{2}+x$`

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) = $8 {x}^{8}+7 {x}^{7}+6 {x}^{6}+5 {x}^{5}+4 {x}^{4}+3 {x}^{3}+2 {x}^{2}+x$`

Calculate the summation series for $x=2$.

`G(2)`
`ans = $3586$`

## Input Arguments

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

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`.

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

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

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### Definite Sum

The definite sum of a series is defined as

`$\sum _{k=a}^{b}{x}_{k}={x}_{a}+{x}_{a+1}+\dots +{x}_{b}.$`

### Indefinite Sum

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

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

such that

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

## Version History

Introduced before R2006a