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# Series

 Series[f, {x, x0, n}]generates a power series expansion for f about the point x=x0 to order (x-x0)n. Series[f, {x, x0, nx}, {y, y0, ny}, ...]successively finds series expansions with respect to x, then y, etc.
• Series can construct standard Taylor series, as well as certain expansions involving negative powers, fractional powers and logarithms.
• Series detects certain essential singularities. On[Series::esss] makes Series generate a message in this case.
• Series can expand about the point x=.
• Series[f, {x, 0, n}] constructs Taylor series for any function f according to the formula .
• Series effectively evaluates partial derivatives using D. It assumes that different variables are independent.
• The result of Series is usually a SeriesData object, which you can manipulate with other functions.
• Normal[series] truncates a power series and converts it to a normal expression.
Power series for the exponential function around :
Convert to a normal expression:
Power series of an arbitrary function around :
In any operation on series, only appropriate terms are kept:
Power series for the exponential function around :
 Out[1]=
Convert to a normal expression:
 Out[2]=

Power series of an arbitrary function around :
 Out[1]=

 Out[1]=
In any operation on series, only appropriate terms are kept:
 Out[2]=
 Scope   (10)
Series can handle fractional powers and logarithms:
Symbolic parameters can often be used:
Laurent series with negative powers can be generated:
Truncate the series to the specified negative power:
Find power series for special functions:
Find the series for a function at a branch point:
With x assumed to be to the left of the branch point, a simpler result is given:
Piecewise functions:
Power series at infinity:
Series can give asymptotic series:
Series expansions of implicit solutions to equations:
Series expansions of unevaluated integrals:
Power series in two variables:
Series is threaded element-wise over lists:
Series generates SeriesData expressions:
Series can work with approximate numbers:
 Options   (4)
Series by default assumes symbolic functions to be analytic:
Use Assumptions to specify regions in the complex plane where expansions should apply:
Without assumptions, piecewise functions appear:
Get expansions in Stokes regions:
 Applications   (8)
Plot successive series approximations to :
Find a series expansion for a standard combinatorial problem:
Find Fibonacci numbers from a generating function:
Find Legendre polynomials by expanding a generating function:
Set up a generating function to enumerate ways to make change using U.S. coins:
The number of ways to make change for \$1:
Find the lowest-order terms in a large polynomial:
Find higher-order terms in Newton's approximation for a root of f[x] near x=a:
Plot the complex zeros for a series approximation to Exp[x]:
Series always only keeps terms up to the specified order:
Operations on series keep only the appropriate terms:
Normal converts to an ordinary polynomial:
Any mathematical function can be applied to a series:
Adding a series of lower order causes the higher-order terms to be dropped:
Differentiate a series:
Solve equations for series coefficients:
Find the list of coefficients in a series:
Use O[x] to force the construction of a series:
ComposeSeries treats a series as a function to apply to another series:
InverseSeries does series reversion to find the series for the inverse function of a series:
When there is an essential singularity, Series will attempt to factor it out:
Numeric values cannot be substituted directly for the expansion variable in a series:
Use Normal to get a normal expression in which the substitution can be done:
Series must be converted to normal expressions before being plotted:
Power series with different expansion points cannot be combined:
Not all series are represented by expressions with head SeriesData:
Some functions cannot be decomposed into series of power-like functions: