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# Tag Archives: Real Analysis

## Zeta logarithmic series

Let denote the zeta function. Prove that

**Solution**

## Mellin transform integral

Evaluate the integral:

**Solution**

We are evaluating the Mellin transform of the function .

where is the Euler’s Gamma function. Hence,

where is the Euler – Mascheroni constant and is the digamma.

## Theta Jacobi limit

Prove that

**Solution**

We are working near . It follows from the Integral Comparison Test that

However,

The result follows from the Sandwich Theorem.

## Double binomial sum

Evaluate the double sum

**Solution**

We sum diagonally , hence:

(1): For it holds .

(2): .

**Conjecture:** Does the following equality

hold?

## Arctan series

Evaluate the series

**Solution**

*One can argue that the same technique used to evaluate the sum here can be used here as well. Unfortunately, this is not the case as the sum does not telescope. However the technique used here is the way to go.*

First of all we note that

However, it is known that

(1)

Let . We note that . Hence,

which is now a matter of calculations. The sum is equal to