## 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.

## An integral with trigonometric and rational function

Evaluate the integral

**Solution**

Let us now consider the function as well as the contour

The function has three simple poles of which only is included within the contour. The residue at turns out to be . Thus

The contribution of the large circle as is whereas the contribution of the small circle as is . This can be seen by parametrising the small circle ( ). Hence:

Using we get that

## A logarithmic integral

Prove that

**Solution**

Now let us consider the complex function where the principal arguement of lies within the interval as well as the contour below

It is clear that has two poles of order at and . The residue at is equal to whereas the residue at is equal to . Thus

Sending and the contribution of both the large and the small circle is . Hence:

Thus the conclusion follows.