Hadamard's n-th root formula

Definition of limsup[edit | edit source]

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 .

Then

Definition 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 12

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 13

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 .
 
 Previous