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.
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.
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
.