A logarithmic integral
Let . Prove that:
Solution
Using the known Fourier series as well as the known generating function of the Catalan numbers
Then, we have successively:
A Gamma summation
Let and
. Prove that
Solution
Consider the function
We easily evaluate that
Hence it follows from Poisson summation formula that
Thus,
The result follows.
A Gudermannian integral
Let . Prove that
where is the Gudermannian function.
Solution
First of all we note that and
. Hence,
An odd trigonometric integral
Let . Prove that
Solution
Since we deduce that
Hence,
since and
. Finally, we note that the sum telescopes hence the result.
A rational harmonic series
Can the rational numbers in the interval be enumerated as a sequence
in such a way that the series
is convergent?