## 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.

## Digamma and Trigamma functions

Let and denote the digamma and trigamma functions respectively. Prove that:

where denotes the Euler – Mascheroni constant.

Solution

We begin with the recently discovered identity:

Letting we get that

Now combining this result here we conclude the exercise.

## Trigamma series

Let denote the trigamma function. Prove that

Solution

## A Riemann sum IV

Using Riemann sums prove that

Solution

Let . Taking logarithms on both sides we get that

Thus the limit follows.

## A Riemann sum III

1. Using Riemann sums evaluate the limit

2. Using the above result prove that

Solution

1. We have successively:

2. Let . Taking logarithms on both sides we get that

The result follows.

### Who is Tolaso?

Find out more at his Encyclopedia Page.