## 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.

## 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

- Using Riemann sums evaluate the limit
- Using the above result prove that

**Solution**

- We have successively:
- Let . Taking logarithms on both sides we get that
The result follows.