Definition of Power Series

9.7 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.

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 and 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.