Equivalent Defintions of

It is well-known that for all complex numbers z,

What follows below is a direct proof of this fact in the case where z is a positive real number. The proof is elementary in that it does not depend on limits superior and inferior, but instead on the Pinching Theorem.

Lemma 198

Suppose that and are positive integers and .Then

Proof: The lemma is clearly true if

since the righthand side expression is never negative. Therefore, suppose that

In this case the proof is by induction on k along with the observation that for , . QED

It follows directly from the Binomial Theorem that for and x > 0

so on the one hand we have

while by applying the Lemma we have

so that

Since it is readily established that

exists for positive by using comparison to a geometric series, we see that

has the same limit as .

Eric S Key
10/24/2000