We think of a power series as a sequence of polynomials

In general, this sequence will converge for certain complex numbers, and diverge for other numbers. A power series determines a function whose domain is the set of all such that converges.

The series
and
are power series that
converge for all
. corresponds to the sequence

and corresponds to

The limits are and , respectively (by definition 11.43.)

Every power series
converges at . (The limit is .)

The series converges only when (see exercise 12.5).

and

we see that in general . We make the convention that

The expression is usually simplified and written without parentheses by use of exercise 3.64:

which is not identical with

but you should be able to see that one series converges if and only if the other does, and that they have the same limits. In the future I will sometimes blur the distinctions between two series like this.

For , let
. Then

If , then and , so by the ratio test, converges absolutely for .

If and , then

so for large , and the series diverges. If , then , so converges by the comparison test, and converges absolutely. This shows that the function

is defined for all , and determines a function from into .

The figure below 12.1 shows the images under of circles of radius for and of rays that divide the disc into twelve equal parts. The images of the interior circles are nice differentiable curves. The image of the boundary circle seems to have interesting properties that I do not know how to demonstrate.

Image of

Let for .

The figure below 12.1 shows the images under of circles of radius for , and of rays that divide the disc into 12 equal parts.

Image of