# Making Power Series Expansions

Series[expr,{x,x_{0},n}] | find the power series expansion of expr about the point to order at most |

Series[expr,{x,x_{0},n_{x}},{y,y_{0},n_{y}}] | |

find series expansions with respect to y, then x |

Functions for creating power series.

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*Mathematica*does not know the series expansion of a particular function, it writes the result symbolically in terms of derivatives.

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In mathematical terms, Series can be viewed as a way of constructing Taylor series for functions.

The standard formula for the Taylor series expansion about the point of a function with derivative is . Whenever this formula applies, it gives the same results as Series. (For common functions, Series nevertheless internally uses somewhat more efficient algorithms.)

Series can also generate some power series that involve fractional and negative powers, not directly covered by the standard Taylor series formula.

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There are, of course, mathematical functions for which no standard power series exist. *Mathematica* recognizes many such cases.

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Especially when negative powers occur, there is some subtlety in exactly how many terms of a particular power series the function Series will generate.

One way to understand what happens is to think of the analogy between power series taken to a certain order, and real numbers taken to a certain precision. Power series are "approximate formulas" in much the same sense as finite-precision real numbers are approximate numbers.

The procedure that Series follows in constructing a power series is largely analogous to the procedure that N follows in constructing a real-number approximation. Both functions effectively start by replacing the smallest pieces of your expression by finite-order, or finite-precision, approximations, and then evaluating the resulting expression. If there are, for example, cancellations, this procedure may give a final result whose order or precision is less than the order or precision that you originally asked for. Like N, however, Series has some ability to retry its computations so as to get results to the order you ask for. In cases where it does not succeed, you can usually still get results to a particular order by asking for a higher order than you need.

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When you make a power series expansion in a variable x, *Mathematica* assumes that all objects that do not explicitly contain x are in fact independent of x. Series thus does partial derivatives (effectively using D) to build up Taylor series.

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You can use Series to generate power series in a sequence of different variables. Series works like Integrate, Sum, and so on, and expands first with respect to the last variable you specify.

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