Let be positive integers. Calculate
where is a non negative number and represents the greatest integer less than or equal to .
Solution
Using the identity we note that
Hence,
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Let be positive integers. Calculate
where is a non negative number and represents the greatest integer less than or equal to .
Solution
Using the identity we note that
Hence,
Let and . Prove that
Solution
Consider the function
We easily evaluate that
Hence it follows from Poisson summation formula that
Thus,
The result follows.
Let . Prove that
where is the Gudermannian function.
Solution
First of all we note that and . Hence,
Let denote the Riemann zeta function. Prove that
where is the generalized harmonic number of order .
Solution
We have successively
Let denote the -th harmonic number and the -th harmonic number of order , namely . Prove that
Solution
Lemma: Let be a sequence such tht . Then
Proof: We have successively:
Thus,
Hence for the original problem if we let then