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Peculiar alternating series
Prove that
Solution
We simply note that
Hyperbolic series
Evaluate the series
Solution
We have successively
since by the unique factorization theorem, any positive even integer can be written in a unique way as
with
and
.
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.
Factorial series
Prove that
Solution
We have successively: