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SeriesCoefficient

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

SeriesCoefficient[f, {x, x0, k}]
    gives the coefficient of (x - x0)^k in the power-series expansion of f
    about x = x0. Works for a concrete integer index k and a finite expansion
point, for any f that Series can expand. HoldAll, Protected.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= SeriesCoefficient[BesselJ[0, x], {x, 0, 4}]
Out[1]= 1/64

In[2]:= SeriesCoefficient[Exp[x], {x, 0, 5}]
Out[2]= 1/120

Implementation notes

  • HoldAll, Protected.
  • Computed by expanding with Series and extracting the k-th coefficient from

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= SeriesCoefficient[Exp[x], {x, 0, 10}]
Out[1]= 1/3628800

The coefficient of x^7 in Tan[x] matches the corresponding tangent number:

In[1]:= SeriesCoefficient[Tan[x], {x, 0, 7}]
Out[1]= 17/315

The coefficient of x^n in 1/(1 - x - x^2) is the n-th Fibonacci number; here F(10) = 89:

In[1]:= SeriesCoefficient[1/(1 - x - x^2), {x, 0, 10}]
Out[1]= 89

In[1]:= SeriesCoefficient[Cos[x], {x, 0, 8}]
Out[1]= 1/40320

Notes

SeriesCoefficient[f, {x, x0, k}] returns the coefficient of (x - x0)^k in the power-series expansion of f about x = x0, for any f that Series can expand and a concrete integer index k. It is computed by expanding f to order k and extracting the single coefficient, so the result is exact (rational or symbolic). SeriesCoefficient is HoldAll, so the expansion variable is held unevaluated.