## An integral with Gamma and digamma

Let denote the Euler’s Gamma function and denote the digamma function. Evaluate

**Solution**

The exercise can also be found at Aops.com .

## An inequality involving harmonic number

Let denote the – th harmonic number and let . Prove that

**Solution**

We might begin with the integral representation of the harmonic number, namely the equation:

So we have to prove the equivelant inequality

and this is obvious using the Cauchy – Schwarz inequality.

The exercise can also be found here .

## A limit

Let denote the -th harmonic number. Evaluate the limit

**Solution**

and of course we have the series represantation of the constant, namely this:

Hence:

Thus

We conclude that the desired limit is just .

## A differential equation

Let be a differential equation such that

Find an explicit formula of .

**Solution**

Making use of the initial condition we get that . Thus . This also means that has no roots in the domain given. Hence the initial condition gives us and the function follows to be .

## Does the trigonometric series converge?

Examine if the series

converges.

**Solution**

is an integer. Let and let be the distance from to the nearest integer. That is

It is immediate that can be expanded periodically with period . Since is an integer we can get that

Thus:

and the conclusion follows.