SeriesCoefficient
Returns a specific coefficient from a Taylor series.
$ wo 'SeriesCoefficient[Sin[x], {x, 0, 3}]'
-1/6
The {x, x0, n} spec also works on an already-computed series.
$ wo 'SeriesCoefficient[Series[Exp[x], {x, 0, 10}], {x, 0, 5}]'
1/120
A symbolic index returns the general term as a Piecewise.
$ wo 'SeriesCoefficient[Exp[x], {x, 0, n}]'
Piecewise[{{n!^(-1), n >= 0}}, 0]
$ wo 'SeriesCoefficient[1/(1 - 2 x), {x, 0, n}]'
Piecewise[{{2^n, n >= 0}}, 0]
A binomial power gives the binomial-series term.
$ wo 'SeriesCoefficient[(1 + x)^5, {x, 0, n}]'
Piecewise[{{Binomial[5, n], Inequality[0, LessEqual, n, LessEqual, 5]}}, 0]
$ wo 'SeriesCoefficient[Log[1 + x], {x, 0, n}]'
Piecewise[{{-((-1)^n/n), n >= 1}}, 0]