Calculus BC: Series: Power Series

Previous Next

We now investigate the convergence of power series, of which the Taylor series we will encounter in the next SparkNote are a special case. A power series is a series of the form

a_nxⁿ

where the a_n are constants and x is a variable. In order to ask if a power series converges, we must first specify the value of x. The n-th partial sum of such a series looks like

a₀ + a₁x + ^... + a_nxⁿ

a polynomial of degree n in the variable x.

One example of a power series is the series

where n! = n(n - 1)(n - 2)^...(2)(1). This series converges absolutely for all values of x. To show this, we use the ratio test. Letting a_n = | x|ⁿ/n!, we have

Now | x|/(n + 1) < 1/2 for large enough n, so (disregarding as many initial terms of the sequence as necessary) we see that the sequence converges.

We now state the key result concerning the convergence of power series. For each power series a_nxⁿ, there exists a radius of convergence, r, which may be either a nonnegative number or infinity. If r is a number, then a_nxⁿ converges absolutely whenever 0≤| x| < r and does not converge absolutely whenever r < | x|. If r = ∞, then a_nxⁿ converges absolutely for all real numbers x.

This fact allows us to define a function on the interval (- r, r) by setting

f (x) =

a_nxⁿ

for all x with | x| < r. Let us return, by way of illustration, to the power series xⁿ/n! introduced earlier. There is no reason why the index must begin at 1. We may as well let it start at 0 (recalling that 0! = 1)--the radius of convergence will still be ∞. The function thus defined for all real numbers,