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Power Series

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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.