power series

in ascending powers of x, with coefficients a0, a1, a2,…, is a power series in x. For example, the geometric series 1 + x + x2 + … + xⁿ + … is a power series; this has a sum to infinity (see series) only if −1 < x < 1 when considered as a real power series or for |x|<1 when considered as a complex power series.

Real and complex power series converge (that is, have a sum to infinity) within a circle of convergence.

The coefficients ai may more generally be taken from a field or commutative ring R. The formal power series in an indeterminate x with coefficients in R themselves form a ring denoted R[[x]]. These power series may be naturally added and multiplied without reference to convergence.

For further examples of power series, see Appendix 11. See also Taylor series.

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