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.