It is time to consider functions other than polynomials. In particular it is time to give a precise description of functions like e^{x},sin
Definition 11.1.1 Let
 (11.1) 
is called a Taylor series centered at a. This is also called a power series centered at a. It is understood that x and a ∈ ℝ. More generally, these variables will be complex numbers, but in this book, only real numbers.
In the above definition, x is a variable. Thus you can put in various values of x and ask whether the resulting series of numbers converges. Defining D to be the set of all values of x such that the resulting series does converge, define a new function f defined on D having values in ℝ as

This might be a totally new function, one which has no name. Nevertheless, much can be said about such functions. The following lemma is fundamental in considering the form of D which always turns out to be of the form B
Proof: It is clear n^{1∕n} ≥ 1. Let n^{1∕n} = 1 + e_{n} where 0 ≤ e_{n}. Then raising both sides to the n^{th} power for n > 1 and using the binomial theorem,

From this the desired result follows because

Theorem 11.1.3 Let ∑ _{k=0}^{∞}a_{k}
Proof: See Definition 3.2.12 for the notion of limsup and liminf. Note

Then by the root test, the series converges absolutely if

and diverges if

Thus define

Next let λ be as described. Then if

It follows that for all k large enough and such x,
Note that the radius of convergence r is given by

Definition 11.1.4 The number in the above theorem is called the radius of convergence and the set on which convergence takes place is called the disc of convergence. Since this book only considers functions of one real variable, it will be called the interval of convergence.
Now the theorem was proved using the root test but often you use the ratio test to find the interval of convergence. This kind of thing is typical in math so get used to it. The proof of a theorem does not always yield a way to find the thing the theorem speaks about. The above is an existence theorem. There exists an interval of convergence from the above theorem. You find it in specific cases any way that is most convenient.
Use Corollary 10.4.9.

because lim_{n→∞}
What of the other numbers z satisfying
Example 11.1.6 Find the radius of convergence of ∑ _{n=1}^{∞}
Apply the ratio test. Taking the ratio of the absolute values of the

Therefore the series converges absolutely if

To see this is the case, the limit, if it exists, is the same as

from an application of L’Hopital’s rule.