# Summation of Infinite Series

There are three functions for evaluating infinite sums of real numbers. The sum should be specified as a map $m$ from the integers to the real field, such that $m(n)$ is the $n^{\rm th}$ term of the sum. The summation begins at term $i$. The precision of the result will be the default precision of the real field.

## `InfiniteSum(m, i): Map, RngIntElt -> FldReElt`

An approximation to the infinite sum $m(i)+m(i+1)+m(i+2)+\cdots$. This function also works for maps to the complex field.

## `PositiveSum(m, i): Map, RngIntElt -> FldReElt`

An approximation to the infinite sum $m(i)+m(i+1)+m(i+2)+\cdots$. Designed for series in which every term is positive, it uses van Wijngaarden’s trick for converting the series into an alternating one. Due to the stopping criterion, terms equal to 0 will create problems and should be removed.

## `AlternatingSum(m, i): Map, RngIntElt -> FldReElt`

```magma
Al: MonStgElt                    Default: "Villegas"
```

An approximation to the infinite sum $m(i)+m(i+1)+m(i+2)+\cdots$. Designed for series in which the terms alternate in sign. The optional argument `Al` can be used to specify the algorithm used. The possible values are `"Villegas"` (the default), and `"EulerVanWijngaarden"`. Due to the stopping criterion, terms equal to 0 will create problems and should be removed.