Series Expansion of an Expression

To expand a function or expression in a Taylor or Laurent series around 0, select the variable around which you want to expand, and either:

• Type [Ctrl][Shift][.] and the "series" keyword followed by a comma and the variable or expression on which to expand, or use the Symbolics toolbar to modify the live symbolics operator.

  – or –

• Choose Variable > Expand to Series from the Symbolics menu.

If you use the menus, you are prompted for an order of expansion. Choose an order (the default is O = 6, the maximum is 99) and click "OK." The "series" keyword generates a default series of order 6. To specify a different order, follow the variable of expansion with a comma and the appropriate integer. For order 6, the highest powered term in the resulting series is of order 5, with a term like O (x⁶) at the end, indicating that the approximation error of the series is bounded by a constant times x⁶ as x tends to 0.

Notes:

• Taylor series (series in positive powers of the variable) are applied to functions that are analytic at 0. Laurent series are applied to functions that have a pole of finite order at 0.
• To expand about a point other than zero, specify the variable value after the "series" keyword using the logical equals. For example:

  x+y series,x =2 →

  expands around the point x = 2.

• To expand a series around more than one variable, follow the "series" keyword with a list of comma-separated variables.
• Taylor series for a function may converge only in some small interval around the center. Furthermore, functions like sin or exp have series with infinitely many terms (how many depends on the order you select). When you approximate a function by the polynomial returned by Expand to Series, the approximation is reasonably accurate close to the center, but may be quite inaccurate for values far from the center.

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