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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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The composition is a metric

Let d be a metric , f be a strictly increasing function and concave on [0, +\infty) such that f(0)=0. Prove that f \circ d is a metric.

Symmetry of Euler sums

Let \zeta denote the Riemann zeta function. Prove that

    \[\sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(k)}}{n^k} = \frac{\zeta(2k)+ \zeta^2(k)}{2}\]

where \mathcal{H}_n^{(s)} is the generalized harmonic number of order s.

Solution

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A limit

Let f:(0, +\infty) \rightarrow \mathbb{R}. If the line y=\lambda x + \beta \; , \; \lambda>0 is an oblique asymptote of \mathcal{C}_f at +\infty then evaluate the limit

    \[\ell = \lim_{x \rightarrow +\infty} \left( f(x) \ln (x+1) - f(x+1) \ln x \right)\]

An integral

Let n \in \mathbb{Z}_{\geq 0}. Evaluate the integral

    \[\mathcal{J}_n = \int_0^\pi \frac{1- \cos nx}{1-\cos x}\, \mathrm{d}x\]

Solution

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