Hadamard's n-th root formula

(Minor edit to remove reference to a course at UCL)
Line 8: Line 8:
that <math>M_N \leqslant M_{N-1}</math> for all <math>N</math>, so we have a non-increasing
that <math>M_N \leqslant M_{N-1}</math> for all <math>N</math>, so we have a non-increasing
sequence of real numbers unless <math>M_N = +\infty</math> for all <math>N</math>.
sequence of real numbers unless <math>M_N = +\infty</math> for all <math>N</math>.
With the above definitions,  
With the above definitions,  

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


From the definition, there exists so that

For such ,
and so for all as required.


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


for the radius of convergence.



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.


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 .


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 .