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 n x n|
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
|a 0 + a 1 x + ... + a n x n|
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.