## Binomial sum

Let be positive numbers with . Prove that

Solution

Using the exercise here we have that

Hence,

## Logarithmic inequality

Let . Prove that

Solution

Let and . Thus,

Thus,

The result follows.

## Contour integral

Evaluate the integral

Solution

The function is meromorphic on . Its only pole is of order . Hence,

Therefore,

## Nested binomial sum

Prove that

Solution

We may begin with the beta function identity for non negative integer values of .

Hence, for non-negative integers

As a result we may compute the nested summation as,

## An arcosine integral

Evaluate the integral