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SeriesData

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SeriesData[x, $x_0$, {$a_0$, $a_1$, ...}, $n_{min}$, $n_{max}$, den]

produces a power series in the variable x about point $x_0$. The ai are the coefficients of the power series. The powers of (x-$x_0$) that appear are $n_{min}$/den, ($n_{min}$+1)/den, ..., $n_{max}$/den.

SeriesData is the Head of expressions generated by Series:

• • series = Series[Cosh[x],{x,0,2}]

• • Head[series]

• • series // FullForm

You can apply certain mathematical operations to SeriesData objects to get new SeriesData objects truncated to the appropriate order.

• • series + Series[Sinh[x],{x,0,3}]

• • Series[f[x],{x,0,2}] * g[w]

  • $$f\left[0\right] g\left[w\right]+g\left[w\right] f^{\prime }\left[0\right] x+\frac{g\left[w\right] f^{\prime \prime }\left[0\right]}{2} x^2+{O\left[x\right]}^3$$

The product of two series on the same neighborhood of the same variable are multiplied:

• • Series[Exp[-a x],{x,0,2}] * Series[Exp[-b x],{x,0,2}]

  • $$1+\left(\mathrm{-}a\mathrm{-}b\right) x+\left(\frac{a^2}{2}+a b+\frac{b^2}{2}\right) x^2+{O\left[x\right]}^3$$

• • D[Series[Exp[-a x],{x,0,2}],a]

  • $$\mathrm{-}x+a x^2+{O\left[x\right]}^3$$

See also Series and SeriesCoefficient.