## An integral

Let . Evaluate the integral

Solution

We note that

This shows that we have a geometric progression. Since , it follows that .

## On a generating function

Let denote the -th harmonic number and the -th harmonic number of order , namely . Prove that

Solution

Lemma: Let be a sequence such tht . Then

Proof: We have successively:

Thus,

Hence for the original problem if we let then

## Harmonic series

Let denote the -th harmonic number. Prove that

Solution

We recall the series . Integrating we get

The last takes the form

where denotes the dilogarithm function. Thus,

## Pell-Lucas series

The Pell-Lucas numbers satisfy and for . Prove that

## summation

Let denote the Euler’s function. Prove that

Solution

The key idea is to rewrite the floor as a sum involving divisors. Hence,