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Peculiar alternating series

Prove that

    \[\sum_{n=1}^{\infty} \frac{1}{n} \left ( \frac{1}{(-1)^n - 5} \right )^n = \frac{1}{2} \ln \frac{16}{21}\]

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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Factorial series

Prove that

    \[\sum_{n=0}^{\infty} \frac{(4n)!}{(4n+4)!} = \frac{\ln 2}{4} - \frac{\pi}{24}\]

Solution

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