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
.
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Mathematica knows the power series expansions for many mathematical functions.
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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.
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Now the square is computed
exactly.
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Applying
Expand gives a result with 11 terms.
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| Series[expr,{x,x0,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.
