Power series supply us with a new way to describe functions: we can specify the coefficients of each power in the series. This raises the following questions.
Given a function defined previously, what does its power series look like?
What do the terms of the power series tell us about the function?
A standard power series looks like
f0 + f1x + f2x2 + ... + fnxn + ...
We can also look at power series for which we subtract some value, call it z, from the variable x. These look like
f0 + f1(x - z) + f2(x - z)2 + ... + fn(x-z)n + ...
A standard notation for such things is
We can answer our second question by differentiating the function represented by the series n times and then setting x = z. Differentiating n times kills off all terms which have degree strictly less than n, and setting x = z kills all terms which have degree strictly greater than n. We are left with the effect of these operations on the nth term alone.
We then can deduce:
f (n)(z) = n! fn
This tells us that the nth term, fn, in this series, is the nth derivative of f(x) evaluated at x =z, divided by n!
This gives us an answer to our first question as well. If we apply this statement to each term in the series, we find:
Let us apply this result to some functions we know. First the exponential function is its own derivative hence its own second derivative and so on. Evaluating all of these at x = z give the same answer, namely exp(z). This tells us:
If we keep differentiating the sine function, we first get cosine then minus sign then minus cosine, then sine again, and the derivatives repeat this pattern in blocks of 4. This gives us:
Exercise 15.4 Do the same thing for cos(x) and do the same for both sine and cosine for z = 0. Deduce the "addition theorems" for sines and cosines from all these results.
Radius of Convergence of Power Series
If you change the magnitude of the series variable, you change the ratio of successive terms. Since series converge when this ratio is any factor fixed factor less than 1, power series typically converge up to some maximum magnitude of the expansion variable, which value is the limiting ratio of . This ratio is called the radius of convergence of the series.
If we define these functions in the complex plane, so that our variables can be complex numbers, this ratio has a geometric meaning. It turns out that it is the distance from the expansion point, here z and the nearest singularity of the function f.
For example, for the geometric series, there is a singularity of the function at x = 1. This means that the radius of convergence, expanding around 0 is 1.
Exercise 15.5 What is the radius of convergence of the functions sine and exp? If you expand the function (1-x)-1 about z = -4, what will the radius of convergence be?