Let be a sequence of real numbers. For each , define
the set obtained by throwing away the first
terms in the
possibly equal to
, it follows
, so we have a non-increasing
sequence of real numbers unless
With the above definitions,
are both possibilities.
If as , then
Suppose if is even but if is odd. Then
of this set is
. The limit as
Intuitively, if is finite, then it is the
highest horizontal asymptote for the sequence (when graphed against ).
Suppose that is a real sequence with
Then given any
From the definition, there exists so that
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.
For the power series Radius of convergence , we have the formula
for the radius of convergence.
Suppose first that
For any positive number
By definition of the limit, it follows that for any given
, and so
This being true for every
, we have
. This is true for every
, the radius of convergence, so finally
We aim to prove the opposite inequality. Suppose is much
smaller than . Then
Lemma, we know that there exists
, this implies
This means that
, and any
. This means that
. Combined with the previous
inequality, we find that
. The cases
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
If has radius of convergence , then the
series has the same radius of
This can be proved by more elementary
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
th root of the coefficient of
in Definition of limsup
By Hadamard's formula, we conclude that the radius of convergence of
Definition of limsup is .