Hadamard's n-th root formula

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We are now ready to prove Hadamard's 'formula' for the radius of
 
We are now ready to prove Hadamard's 'formula' for the radius of
 
convergence of a complex power series.
 
convergence of a complex power series.
 
  
 
{{Theorem|title=4|
 
{{Theorem|title=4|
  For the power series  [[Course:Complex_Analysis_(Intermediate_Level)/Power_Series_Background/Radius_of_covnergence|Radius of convergence]] , we have the
+
  For the power series  [[Course:Complex_Analysis_(Intermediate_Level)/Power_Series_Background/Radius_of_covnergence|Radius of convergence]] , we have the formula
  formula
 
 
<math display="block">\frac{1}{r} = \limsup_{n\to \infty} |a_n|^{1/n}</math>
 
<math display="block">\frac{1}{r} = \limsup_{n\to \infty} |a_n|^{1/n}</math>
 
for the radius of convergence.
 
for the radius of convergence.

Revision as of 14:15, 26 October 2016

Definition of limsup

Let be a sequence of real numbers. For each , define

the set obtained by throwing away the first terms in the sequence. Let
possibly equal to . Since , it follows that for all , so we have a non-increasing sequence of real numbers unless for all .

ThenDefinition (8):

With the above definitions,

where and are both possibilities.


Example: (9)

If as , then .


Example: (10)

Suppose if is even but if is odd. Then

The of this set is . The limit as is .


Intuitively, if is finite, then it is the highest horizontal asymptote for the sequence (when graphed against ).


Lemma (11):

Suppose that is a real sequence with

Then given any there exists such that which


Proof:

From the definition, there exists so that

For such ,
and so for all as required.


Remark:

The first inequality in Definition of limsup shows that for each there exists such that . By taking an increasing sequence of , it follows that for infinitely many .



We are now ready to prove Hadamard's 'formula' for the radius of convergence of a complex power series.


Theorem (4):

For the power series Radius of convergence , we have the formula

for the radius of convergence.


Proof:

Put

Assume that . Suppose first that , so
Taking th roots,
For any positive number , as . By definition of the limit, it follows that for any given , there exists such that implies . Thus for , , and so
This being true for every , we have Hence and . This is true for every , the radius of convergence, so finally
We aim to prove the opposite inequality. Suppose is much smaller than . Then
By the Lemma, we know that there exists so that implies that
If , this implies
This means that for all , and any . This means that . Combined with the previous inequality, we find that . The cases and require slight modifications of the argument, and are left as exercises.


Remark:

The th root test has the advantage that it works always, provided one can calculate the , of course. This in contrast to the ratio test, which only works if exists.




Proposition (14):

If has radius of convergence , then the series has the same radius of

convergence .


Proof:

This can be proved by more elementary means, but we shall prove it as an illustration of Hadamard's formula. We have observed in Remark~ Radius of convergence that the radius of convergence of

is the same as that of
Now the th root of the coefficient of in Definition of limsup is Since , it follows that
By Hadamard's formula, we conclude that the radius of convergence of Definition of limsup is .

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