Let denote the -th harmonic number. Evaluate the limit
and of course we have the series represantation of the constant, namely this:
We conclude that the desired limit is just .
Let be a differential equation such that
Find an explicit formula of .
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 .
Examine if the series
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
and the conclusion follows.
Evaluate the integral
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
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.