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\pi \right) -\sec \left( \pi \right) \csc ^{2}\left( \pi
\right) }{\pi }}$How to prove$\sum_{k=0}^{\infty}{\frac{1}{(k+1)(k+3)}}={\frac {
\pi \csc \left( \pi \right) -\sec \left( \pi \right) \csc
^{2}\left( \pi \right) }{\pi }}$? I tried to make substitutions$t=k+1$and$s=k+3\$ but it resulted in a messy series and I am not sure how to procede. Any help would be appreciated.

A:

Hint: $$\frac1{k+1}=\int_0^\infty t^{k}\,e^{ -t}\,dt,\qquad\frac1{k+3}=\int_0^\infty t^{k+2}\,e^{ -t}\,dt$$
$$\frac1{(k+1)(k+3)}=\int_0^\infty t^k\,e^{ -t}\int_0^t t^{k+2}\,e^{ -t}\,dt\,dt=\int_0^\infty \frac{t^k}2\,dt$$
Can you finish from here?

House of La Bonte

The House of La Bonte is one of the most memorable and best-known French dynasties during the 18th century.

Origins

Jean de La Bonte (or de Blont) was the first of his name to hold the seat of Le Bonnet on the River Saône in the Dauphiné, the smallest of the three departments of the old province of Savoy (the others being the department of Chablais and the department of Piedmont), through which he passed to the Dau
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