## Limit of a sequence

Let be a continuous function and be the set of all positive integers such that there exists such that

Prove that is infinite and evaluate the limit

## A factorial limit

Let denote the factorial of a real number; that is . Evaluate the limit:

**Solution**

It holds that

where denotes the -th harmonic number and the Euler – Mascheroni constant.

## Linear isometry

Let . If:

- for all

then prove that is linear.

**Solution**

For convenience, identify with here. Then note that for any such function , also a solution for any point on the unit circle. Also is a solution. Note that and hence we can wlog assume that . So is a point on the unit circle with distance to . Hence , so w.l.o.g. assume that . But then for any , both and have the same distance to and . So supposing , all lie on the perpendicular bisector between these points and in particular and are collinear which clearly is absurd. Hence for all which proves the claim.

## Zeta logarithmic series

Let denote the zeta function. Prove that

**Solution**

## Mellin transform integral

Evaluate the integral:

**Solution**

We are evaluating the Mellin transform of the function .

where is the Euler’s Gamma function. Hence,

where is the Euler – Mascheroni constant and is the digamma.