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## A root limit

Let be positive real numbers such that . Prove that

Solution

Without loss of generation , let . Then,

since forall . Thus, by the squeeze theorem it follows that

## A limit!

Evaluate the limit:

Solution

Let . Then,

It follows by Stolz–Cesàro that

Hence .

## Integral of Jacobi Theta function

Let denote one of the Jacobi Theta functions. Prove that

Solution

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,

Hence,

and the exercise is complete.

## Logarithmic Gaussian integral

Let denote the Euler’s constant. Prove that:

Solution

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,

## Series of zeta sum

Let denote the Riemann zeta function. Evaluate the series:

Solution

Let . We are proving the more general result.

where denotes the digamma function.

First of all, we note that:

Thus,

due to the reflection formula .

Side note: If then the sum equals .

### Who is Tolaso?

Find out more at his Encyclopedia Page.