# Power Series

The mathematical operations we have discussed so far are *exact*. Given precise input, their results are exact formulas.

In many situations, however, you do not need an exact result. It may be quite sufficient, for example, to find an *approximate* formula that is valid, say, when the quantity is small.

This gives a power series approximation to

for

close to

, up to terms of order

.

Out[1]= | |

*Mathematica* knows the power series expansions for many mathematical functions.

Out[2]= | |

If you give it a function that it does not know,

Series writes out the power series in terms of derivatives.

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Power series are approximate formulas that play much the same role with respect to algebraic expressions as approximate numbers play with respect to numerical expressions. *Mathematica* allows you to perform operations on power series, in all cases maintaining the appropriate order or "degree of precision" for the resulting power series.

Here is a simple power series, accurate to order

.

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When you do operations on a power series, the result is computed only to the appropriate order in

.

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This turns the power series back into an ordinary expression.

Out[6]= | |

Now the square is computed

*exactly*.

Out[7]= | |

Applying

Expand gives a result with 11 terms.

Out[8]= | |

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

Normal[series] | truncate a power series to give an ordinary expression |

Power series operations.