## An integral with Gamma and digamma

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

Solution

We have successively

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

We begin by the simple observation that

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

We consider the function which is clearly constant because

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

The key lies in the fact that the number

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.