# SeriesCoefficient

SeriesCoefficient[series,n]

finds the coefficient of the n -order term in a power series in the form generated by Series.

SeriesCoefficient[f,{x,x0,n}]

finds the coefficient of in the expansion of f about the point .

SeriesCoefficient[f,{x,x0,nx},{y,y0,ny},]

finds a coefficient in a multivariate series.

# Details and Options

• In the form SeriesCoefficient[f,{x,x0,n}], the order n can be symbolic.
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters Method Automatic method to use
• For explicit SeriesData objects, the form SeriesCoefficient[series,{nx,ny,}] can also be used.

# Examples

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## Basic Examples(4)

Find the coefficient for a term in a series:

Find the coefficient of the general term in a series:

Find the coefficient for a term in a multivariate series:

Find the coefficient for a general term in a multivariate series:

## Scope(6)

Compute a series coefficient:

Plot the resulting sequence:

Rational functions:

Elementary functions:

Special functions:

In general a DifferenceRoot function may be required to express the solution:

Find the coefficients in multivariate functions:

## Options(3)

### Assumptions(2)

Coefficients of the expansion of the Chebyshev polynomials:

Use Assumptions to get a simpler result:

With no Assumptions, general results are generated:

With Assumptions a result valid under the given assumptions is given:

### Method(1)

This generates a DifferenceRoot object when possible:

## Applications(4)

Find the Fibonacci number from its generating function:

Find a Chebyshev polynomial from its generating function:

Solve a linear difference equation:

Add the initial value equation and solve the algebraic equation for the transform:

Find the expression for y[n]:

Use RSolve:

Find the coefficient of the general term in the power series expansion of 1/(1+x):

Obtain the power series expansion in Inactive form:

Make a table of the power series expansions for different functions:

## Properties & Relations(4)

Use DiscreteAsymptotic to compute an asymptotic approximation:

The coefficients of a truncated series expansion:

The general coefficient formula:

The general formula agrees with the truncated expansion:

CoefficientList finds all coefficients in a series:

SeriesCoefficient is closely related to InverseZTransform:

## Possible Issues(2)

Series coefficients can be functions of the expansion variable:

General coefficients of series may not be available:

## Neat Examples(2)

Series coefficient for a hypergeometric function:

Create a gallery of common series coefficients:

Wolfram Research (1996), SeriesCoefficient, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesCoefficient.html (updated 2008).

#### Text

Wolfram Research (1996), SeriesCoefficient, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesCoefficient.html (updated 2008).

#### CMS

Wolfram Language. 1996. "SeriesCoefficient." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/SeriesCoefficient.html.

#### APA

Wolfram Language. (1996). SeriesCoefficient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SeriesCoefficient.html

#### BibTeX

@misc{reference.wolfram_2024_seriescoefficient, author="Wolfram Research", title="{SeriesCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/SeriesCoefficient.html}", note=[Accessed: 20-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_seriescoefficient, organization={Wolfram Research}, title={SeriesCoefficient}, year={2008}, url={https://reference.wolfram.com/language/ref/SeriesCoefficient.html}, note=[Accessed: 20-September-2024 ]}