8. Sequences of Functions
8.3. Series and Power Series
This section will combine two theories we discussed previously:
 sequences of functions that could converge pointwise or uniformly
 numeric series defined as a limit of a (numeric) sequence of partial sums
A simple example for a numeric series was the geometric series . If a numeric series converged, it represented a complicated way to express the resulting limit value. For example, a complicated way to write the number 1 would be:
1 = ^{1}/_{2} + ^{1}/_{4} + ^{1}/_{8} + ^{1}/_{16} + ^{1}/_{32} + ... =
In this section we will replace the numeric summand by one that depends on x and try to analyze what happens.
Definition 8.3.1: A Function Series  
Suppose { f_{n}(x) } is a sequence of functions and we
define the Nth partial sum as
S_{N}(x) = f_{n}(x)Let D be the set of points for which the sequence of partial sums converges pointwise. Then, for x D, we denote the resulting limit function by f_{n}(x) = S_{N}(x) = f_{n}(x) 
It is clear that each S_{N}, being a finite sum, is a welldefined function that inherits its properties from the f_{n}'s. But we would like to know when the infinite sum is welldefined and what can be said about its properties.
The above examples (with different degrees of difficulty) show that function series can result in simple functions that share properties with the individual terms f_{n}(x) yet can be more complicated at the same time. For example, each term in the geometric function series is differentiable for all x yet when we add up infinitely many of these simple terms the resulting function is not even defined for x > 1.
We would like to develop a general theory of function series that will allow us to find properties of series quickly and efficiently. The basis of our discussion will be:
Theorem 8.3.3: Weierstrass Convergence Theorem  
Suppose f_{n} are a sequence of functions defined on
D such that
 f_{n} _{D} <where  f_{n} _{D} is the supnorm on D. Then the (function) series f_{n}(x) converges absolutely and uniformly on D to a function f. If, in addition, each f_{n} is continuous, the limit function f is also continuous on D. 
Weierstrass' theorem, if it applies, is nice because it lets you draw conclusions about a series of functions by looking at a numeric series of sup's. A simple example is the geometric series, writen as a series of monomials in x, each defined on a closed subset of (1, 1):
Example 8.3.4: Function Series Examples  

The above examples, especially the first, might seem somewhat special and perhaps a little contrived. After all, in the first example the terms of the series are so simple and add up (in the proof) so nicely that there might be reason to suspect an example readymade for students. The other series, on the other hand, seem much more complicated than its constituent components. As it turns out, adding up a bunch of relatively simple functions, even monomials, can yield very complicated functions. Here it is definitely true:
The sum can be much more complicated than its parts!
The next theorem is going to be the cornerstone of a new theory that will even (finally and unexpectedly) provide a solid theoretical foundation for our trig functions sin and cos. Sic!
But, let's get started, as usual with a new definition:
Definition 3.3.5: Power Series  
A function series of the form
a_{n} (x  c)^{n} = a_{0} + a_{1}(xc) + a_{2}(xc)^{2} + ...is called a (formal) power series centered at c. 
In other words, a power series is an infinite series of functions where each term consists of a coefficient a_{n} and a power (xc)^{n}. Here are a few examples of power series:
Example 3.3.6: Formal Power Series Examples  

The above definition of a power series is "formal" because the series may or may not converge. But power series, actually, have very nice and structured convergence properties:
Theorem 3.3.7: Power Series  
Every power series
a_{n} (x  c)^{n} =
a_{0} +
a_{1}(xc) +
a_{0}(xc)^{2} + ...
centered at c has the following properties:
Note that it is possible for the radius of convergence to be zero (i.e. the power series converges only for x = c) or to be (i.e. the series converges for all x). Note that this theorem is sometimes called Abel's theorem on Power Series. 
A power series is, simply put, an "infinite polynomial", i.e. a polynomial of "infinite" degree:
 p(x) = 1 + 2x + 3x^{2} + 4x^{3} is a polynomial of degree 3
 q(x) = 1 + 2(x1) + 3(x1)^{2} is a polynomial of degree 2 centered at c = 1. Note that we could work out the parenthesis to obtain an equivalent polynomial centered at c = 0 (do it)
 f(x) = 1 + 2x + 3x^{2} + 4x^{3} + ... = n x^{n1} is a power series centered at c = 0 with some radius of convergence
 g(x) = 1 + 1/2(x+2) + 1/4(x+2)^{2} + 1/8(x+2)^{3} + ... = 1/2^{n} (x+2)^{n} is a power series centered at c = 2 with some radius of convergence
A power series converges when x = c because then all but the first term are zero. In addition, the series converges in the open interval c  r to c + r and diverges outside that interval, according to the theorem. The theorem makes no statement about convergence or divergene at the endpoints c  r and c + r, so these endpoints should be investigated 'manually'.
If you draw a circle centered at c with radius r then the circle intersects the xaxis at the points c  r and c + r and thus defines the regions of convergence and divergence. But it turns out that power series are best defined in the space of complex numbers, i.e. all coefficients and variables are allowed to be complexvalued. The Power Series theorem remains true in the complex plane and the set z  c < r would be a true circle inside which the (complex) series converges and diverges outside. On the boundary of the disk no general statement exists except for the fact that there must be at least one (possibly complex) point on the boundary where the series does not converge (otherwise you could increase the radius of convergence slightly).
Power series are studied extensively in Complex Analysis; in real analysis they are only a shadow of their true self ...
As a final note, the above theorem gives an explicit formula for the radius of convergence for a power series, which is nice. However, in most cases we could also apply our familiar Ratio Test we studied in chapter 4 to the terms of the series including the powers of x and to 'solve' for x. That way there is one less formula to memorize (assuming of course you remember the ratio test, which surely you do :)!
For example, consider the power series . Clearly the series has c = 2 as center of convergence. To find the radius of convergence, we could apply the above formula:
r = lim sup  a_{n} / a_{n+1}
where a_{n} = 3n / 2^{n}. We get:
r =
=
Thus, the radius of convergence is 2 and the series converges absolutely and uniformly on any subinterval of x  2 < 2.
To apply the ratio test to the same series, recall that the ratio test says that a_{n} converges absolutely if lim sup a_{n+1} / a_{n} < 1. Note that for the ratio test the a_{n+1} will be on top and will include powers of x whereas in the formula for the radius of convergence it shows up on the bottom and does not include any x's.
Let's now apply the ratio test to our series, but this time we let a_{n} = 3n / 2^{n} (x2)^{n}. According to the ratio test our series converges absolutely if:
But that simplifies to the condition
As before, this means that the series converges if x  2 < 2, i.e. the radius of convergence is 2.
Either method has pros and cons:
 If you apply the formula for the radius of convergence directly, the terms seem a little simpler since you don't carry around any powers of x. On the other hand, you need to remember to put the term a_{n} on top, different from the wellknown ratio test.
 If you apply the ratio test you do not need to remember any additional formula. But you need to carry around powers of x (which will, on the other hand always reduce nicely) and you must remember to solve the resulting inequality.
Regardless of the method we used to find the above radius of convergence r=2 we still need to check convergence at the endpoints x=0 and x=4. In this case it would turn out that the series diverges at both endpoints  make sure to verify that.
Example 8.3.8: Power Series Examples  

Occasionally the center of a power series is more tricky to detect, or you might want to 'recenter' a series (which does not always work). The following examples will let you gain some more experience with power series.
Example 8.3.9: Power Series Center  

Of course polynomials are relatively simple functions: they can be added, subtracted, and multiplied (but not divided), and you again get a polynomial. Differentiation and integration are particularly simple and yield again polynomials. Lot's more is known about polynomials (e.g. they can have at most n zeros) and we feel pretty comfortable with them.
As it will turn out, power series share many of these properties, allowing us to think of them as "polynomial with infinite degree". Since we can add, subtract, and multiply absolutely convergent series (see chapter 4) we can add, subtract, and multiply (think Cauchy product) power series, as long as they have overlapping regions of convergence. Even differentiating and integrating works as it should:
Theorem 8.3.10: Differentiating and Integrating Power Series  
Let
a_{n} (x  c)^{n}
be a power series centered at c with radius of convergence
r > 0. Then:

The essence of this theorem, thinking of power series as infinite polynomials, is:
a_{0} + a_{1}(xc) + a_{2}(xc)^{2} + ... dx =
= a_{0} dx + a_{1}(xc) dx + a_{2}(xc)^{2} dx + ... =
= a_{0} (xc) + 1/2 a_{1}(xc)^{2} + 1/3 a_{2}(xc)^{3} + ... + const
and for differentiation:
a_{0} + a_{1}(xc) + a_{2}(xc)^{2} + ... =
= a_{0} + a_{1}(xc) + a_{2}(xc)^{2} + ... =
= a_{1} + 2 a_{2}(xc) + 3 a_{3}(xc)^{2} + ...
Of course the derivative of a power series is again a power series with the same center and radius of convergence as the original series. Thus, a power series can be differentiated again, and again, and again, so that we have the following corrolary:
Corollary 8.3.11: Power Series is infinitely often Differentiable  
If a power series f(x) = a_{n} (xc)^{n} has radius of convergence r, then f is infinitely often differentiable for xc < r. In other words, f C(cr, c+r) . 
The next examples let you experiment with differentiating and integrating power series. You will find that sometimes power series represent wellknown "simple" functions, a theme we will pick up in detail in the next section.
Example 8.3.12: Differentiating and Integrating Power Series  

Example 8.3.13: A Series Approximation of Pi  
