Power Series or Function

We have now seen that when a power series is used to define a function, then that function is very well behaved, and we can manipulate it by manipulating, in the obvious way, the corresponding power series. However there are snags. A function has one definition which works everywhere it makes sense (at least for simple functions), whereas the power series corresponding to a function depends also on the point about which the expansion is happening. An example will probably make this clearer than further discussion.

Example 7.14   Give the power series expansions for the function f (x) = $${\frac{{1}}{{1-x}}}$$.

Solution. We can already do this about 0 by the Binomial Theorem; we have:

$${\frac{{1}}{{1-x}}}$$ = 1 + x + x² + x³ +...+ xⁿ +...    for | x| < 1.    

To expand about a different point, e.g. about 3, write y = x - 3. Then

$${\frac{{1}}{{1-x}}}$$ = $${\frac{{1}}{{1- (y+3)}}}$$ = $${\frac{{1}}{{-2-y}}}$$ = - $${\frac{{1}}{{2}}}$$.$${\frac{{1}}{{1-y/2}}}$$,

and again using the Binomial Theorem on the last representation, we have

$${\frac{{1}}{{1-x}}}$$ = - $${\frac{{1}}{{2}}}$$$$\left(\vphantom{1 + y/2 + y^2/4 + y^3/9 + \ldots + y^n/2^n + \ldots }\right.$$1 + y/2 + y²/4 + y³/9 +...+ yⁿ/2ⁿ +...$$\left.\vphantom{1 + y/2 + y^2/4 + y^3/9 + \ldots + y^n/2^n + \ldots }\right)$$    for | y/2| < 1.    

Writing this in terms of x gives

$${\frac{{1}}{{1-x}}}$$ = - $${\frac{{1}}{{2}}}$$$$\left(\vphantom{1 + \frac{x-3}{2} + \frac{(x-3)^2}{4} + \ldots + \frac{(x-3)^n}{2^n} + \ldots }\right.$$1 + $${\frac{{x-3}}{{2}}}$$ + $${\frac{{(x-3)^2}}{{4}}}$$ +...+ $${\frac{{(x-3)^n}}{{2^n}}}$$ +...$$\left.\vphantom{1 + \frac{x-3}{2} + \frac{(x-3)^2}{4} + \ldots + \frac{(x-3)^n}{2^n} + \ldots }\right)$$    for | x - 3| < 2.    

It should be no surprise that this is the Taylor series for the same function about the point 3. And it is in fact not an accident that the radius of convergence of the new series is 2. More investigation (quite a lot more - mainly for complex functions) shows the radius of convergence is always that of the largest circle that can be fitted into the domain of definition of the function. And that is why it is of interest to sometimes consider power series as complex power series. The power series expansion for (1 + x²)^-1 has radius of convergence 1. This seems implausible viewed with real spectacles, but totally explicable when it is realised that the two points i and - i are stopping the expansion from being valid in a larger circle.