## A hypergeometric series

Let such that . Evaluate the series:

**Solution**

**Lemma 1: **For the functions , it holds that:

**Lemma 2: **It holds that:

*Proof: *Simple calculations using Lemma reveal the identity.

**Lemma 3: **Using Lemma 2 it holds that

and as a consequence

Then, successively we have that:

## A square root limit

Let . Prove that

**Solution**

It holds that

Thus,

**Note:** Similarly, it holds that .

## Searching for the …function

Find all functions such that and

**Solution**

First of all we note that

Thus,

Since the integrand is positive it only remains that

Setting and at the last equation we have that:

Since is positive we conclude that

which satisfies the given conditions.

## Inequality of a concave function

Let be a concave function. Prove that

**Solution**

Since is concave , it holds that

By setting and we get that . Thus,

and the exercise is complete.

## Convergent sequence

Let be a sequence of positive real numbers such that

(1)

Prove that converges.

**Solution**

Fix and let . Then, there exist such that where . Thus,

Letting it follows that

Since this holds forall it follows that and the result follows.