## Factorial series

Prove that

**Solution**

We have successively:

## The composition is a metric

Let be a metric , be a strictly increasing function and concave on such that . Prove that is a metric.

## Symmetry of Euler sums

Let denote the Riemann zeta function. Prove that

where is the generalized harmonic number of order .

**Solution**

We have successively

## A limit

Let . If the line is an oblique asymptote of at then evaluate the limit

## An integral

Let . Evaluate the integral

**Solution**

We note that

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