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Limit of geometric mean of binomial coefficients

Let denote the geometric mean of the binomial coefficients

Prove that .

Solution

We note that

On the other hand the following lemma holds:

Lemma: Let be a monotonic function. It holds that

Proof: Due to monotony it holds that for . Hence summing over all these values of k we get that

The result follows.

Applying the above to on we get that:

Thus,

(1)

Similarly, applying the above to on we get that:

(2)

The result follows.