Let be positive real numbers such that . Prove that
Without loss of generation , let . Then,
since forall . Thus, by the squeeze theorem it follows that
Evaluate the limit:
Let . Then,
It follows by Stolz–Cesàro that
Let denote one of the Jacobi Theta functions. Prove that
We have successively,
The sum is evaluated as follows. Consider the function
and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,
and the exercise is complete.
Let denote the Euler’s constant. Prove that:
First of all by making the substitution we get that
For let us consider the function
where is the Euler’s Gamma function. Differentiating once we get:
Thus, the desired integral is obtained by setting . Hence,
Let denote the Riemann zeta function. Evaluate the series:
Let . We are proving the more general result.
where denotes the digamma function.
First of all, we note that:
due to the reflection formula .
Side note: If then the sum equals .