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Definition of Power Series


So far we have studied series of constants,


One can also form a series of functions


Such a series will converge for some values of x and diverge for others. The sum of the series is a new function


which is defined at each point x0 where the series converges. We shall concentrate on a particular kind of series of functions called a power series. Its importance will be evident in the next section where we show that many familiar functions are sums of power series.


A power series in x is a series of functions of the form


The nth finite partial sum of a power series is just a polynomial of degree n,


The infinite partial sums are polynomials of infinite degree,


At x = 0 every power series converges absolutely,


(In a power series we use the convention a0x0 = a0.) If a power series converges absolutely at x = u, it also converges absolutely at x = -u, because the absolute value series 09_infinite_series-429.gif and 09_infinite_series-430.gif are the same.

Intuitively, the smaller the absolute value |x|, the more likely the power series is to converge at x. This intuition is borne out in the following theorem.

Last Update: 2006-11-08