# Documentation

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# `ORDER`

Default number of terms in series expansions

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## Description

The environment variable `ORDER` controls the default number of terms that the system returns when you compute a series expansion.

Possible values: Positive integer less than 231. The default value is 6.

The functions `taylor`, `series`, and `asympt` have an optional third argument specifying the desired number of terms of the requested series expansion, counting from the dominant term on (relative order). If this optional argument is missing, then the value of `ORDER` is used instead.

`ORDER` may also affect the results returned by the function `limit`.

Deletion via the statement “`delete ORDER`” resets `ORDER` to its default value 6. Executing the function `reset` also restores the default value.

In some cases, the number of terms returned by `taylor`, `series`, or `asympt` may not agree with the value of `ORDER`. Cf. Example 2.

## Examples

### Example 1

In the following example, we compute the first 6 terms of the series expansion of the function `exp(x)/x^2` around the origin:

`series(exp(x)/x^2, x = 0)`

To obtain the first 10 terms, we specify the third argument of `series`:

`series(exp(x)/x^2, x = 0, 10)`

Alternatively, we increase the value of `ORDER`. This affects all subsequent calls to `series` or any other function returning a series expansion:

`ORDER := 10: series(exp(x)/x^2, x = 0)`

`taylor(x^2/(1 - x), x = 0)`

Finally, we reset `ORDER` to its default value 6:

`delete ORDER: taylor(x^2/(1 - x), x = 0)`

### Example 2

The number of terms returned by `series` may differ from the value of `ORDER` when cancellation or rational exponents occur:

`ORDER := 3:`
`series(exp(x) - 1 - x - x^2/2 - x^3/6, x = 0)`

`series(1/(1 - sqrt(x)), x = 0)`

`delete ORDER:`