Examples for

Series Expansions

Wolfram|Alpha can compute Taylor, Maclaurin, Laurent, Puiseux and other series expansions. A series expansion is a representation of a mathematical expression in terms of one of the variables, often using the derivative of the expression to compute successive terms in the series. A partial sum of a series expansion can be used to approximate a math expression numerically.
Taylor Series

Analyze a function using the Taylor power series.

Find a Taylor series expansion:

taylor series sin x

Expand around a specified point:

series sin x at x=pi/4

Specify the order of the expansion:

series sin x to order 20

Specify the center point and the order of the expansion:

series (sin x)/(x-pi) at x=pi to order 10
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Laurent Series

Represent a function as a Laurent series.

Find a Laurent series expansion:

series cot z
series (sin z)/z^3 to order 10
series exp(1/x) at x = infinity
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Puiseux Series

Use a power series with fractional exponents to approximate a function.

Find a Puiseux series expansion:

series sqrt(sin x) at x=0
series exp(sqrt(x))

Find a generalized Puiseux series expansion:

series log(x) cos(x)
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