Radius of covnergence

Let

be a power series centred at . Here the , the of the power series are fixed complex numbers.


Proposition 4

There exists , , such that Radius of convergence is absolutely convergent if and divergent if . The convergence is moreover on any smaller closed disc , ().

 


Proof

For any let

where we allow the possibility if the sequence is unbounded. We note that if , then (where this may have to be interpreted as ). To show this, suppose for example that . Then
Thus every element of the sequence is and so the same is true of their sup . A similar argument shows that if and , then also .

Since , it makes sense to ask for the largest with . In other words, define

The claim is that has the required properties of the Proposition. (Note that can be either finite or , depending on the particular power series.)

If , then there exists with . We have seen that is finite, and so

The absolute convergence follows by comparision with the geometric series which is convergent because .

If , then so that the sequence is unbounded. Thus the terms in Radius of convergence don't tend to zero and the partial sums cannot converge.

Finally, let be a closed subdisc. Let be chosen so that . Then we can run the above argument uniformly for all (even ),

if . Uniform convergence on is equivalent to the following version of Cauchy's criterion:

Given , there exists (depending upon such that if , then for all .

This is satisfied in our case for because

and this tends to zero as if . (In the last inequality in Radius of convergence we have bounded a finite geometric progression by its sum to infinity. This is a device that is often handy.)

 


Remark 5

It follows from the definition that if Radius of convergence has radius of convergence , then so do the related series

where is an integer, is an integer, and we assume if is negative.

Thus the radius of convergence is unaffected by throwing away any finite number of terms in the sum, or by multiplying by a fixed power of .

 


Remark 6

Consider the three series

In each case the radius of convergence (exercise). In the case of Radius of convergence and Radius of convergence , defined in Radius of convergence , is finite, in fact equal to . In the case of Radius of convergence , .

We note also that these behave differently at the boundary of the radius of convergence, when . In cases (a) and (b), the series diverge for all with . (The terms don't even go to zero.) In case (c), the series converges for all values of with .

Thus you cannot say anything about a power series at its radius of convergence: the behaviour can be arbitrarily wild.

 



Remark 7

I wrote the section on power series before the section on uniform convergence. The arguments may seem quite repetitive. In fact, with the -test in hand, the discussion of power series could have been streamlined by using the -test with for the last part of the proof of Proposition~ Radius of convergence .

 
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