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A finite summation

Let m,n be positive integers. Calculate

    \[\mathcal{S} = \sum_{k=1}^{2n} \prod_{i=0}^{m} \left ( \left \lfloor \frac{k+1}{2} \right \rfloor + \alpha + i \right )^{-1}\]

where \alpha is a non negative number and \left \lfloor x \right \rfloor represents the greatest integer less than or equal to x.

Solution

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Hyperbolic series

Evaluate the series

    \[\mathcal{S} = \sum_{n=1}^{\infty} \frac{1}{\sinh 2^n}\]

Solution

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A logarithmic integral

Let \alpha \in \mathbb{R}. Prove that:

    \[\int_{0}^{\infty} \frac{\arctan \alpha \sin^2 x}{x^2}\, \mathrm{d}x = \frac{\pi}{\sqrt{2}} \cdot \frac{\alpha}{\sqrt{1+ \sqrt{1+\alpha^2}}}\]

Solution

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A Gamma summation

Let a \notin \mathbb{Z} and a > \frac{1}{2}. Prove that

    \[\sum_{n=0}^{\infty} \frac{2}{\Gamma \left ( a + n \right ) \Gamma \left ( a - n \right )} = \frac{2^{2a-2}}{\Gamma \left ( 2a - 1 \right )} + \frac{1}{\Gamma^2 (a)}\]

Solution

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A Gudermannian integral

Let n \in \mathbb{N}. Prove that

    \[\int_{0}^{\infty} \int_{0}^{\infty} \frac{\mathrm{gd}^n \left ( xy \right )}{\cosh xy} \sin y \, \mathrm{d} \left ( x, y \right ) = \frac{1}{n+1} \cdot \left ( \frac{\pi}{2} \right )^{n+2}\]

where \mathrm{gd} is the Gudermannian function.

Solution

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