Power series

A power series centred at is an infinite series with variable terms of the form

where is an infinite sequence of real numbers, and . In the case where it is often just called a power series. The domain of is therefore the set for which the series converges. A variant of the Ratio test for absolute convergence can be used to find the radius of convergence

where

is absolutely convergent on the interval centred at given by
must be checked manually for divergence at each
is divergent for all other

Note this gives rise to the special cases

If then is absolutely convergent for and divergent everywhere else.
If , i.e. the limit diverges, then is absolutely convergent for all .

Another property arising from this is that it is not possible for a power series to be convergent in several separate points or intervals.

Power series are used to define the notion of an analytic function, i.e. the differentiability class. Note the derivative of a power series always has the same radius of convergence.

Examples

Perhaps the most well known power series is the Taylor series centred at , often denoted

which in the case of is called the Maclaurin polynomial. In fact, Borel's theorem states that every power series is in fact a Taylor series of some smooth function.

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