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.
We now state the key result concerning the convergence of power series. For each power series a _{n} x ^{n} , there exists a radius of convergence, r , which may be either a nonnegative number or infinity. If r is a number, then a _{n} x ^{n} converges absolutely whenever 0≤| x| < r and does not converge absolutely whenever r < | x| . If r = ∞ , then a _{n} x ^{n} 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 _{n} x ^{n} |
for all x with | x| < r . Let us return, by way of illustration, to the power series x ^{n}/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,
f (x) = = 1 + x + + + ^{ ... } |
happens to be equal to the exponential function f (x) = e ^{x} . We will return to this intriguing "coincidence" in the next chapter.