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Pell – Lucas series

The Pell – Lucas numbers \mathcal{Q}_n are defined as follows \mathcal{Q}_0 = \mathcal{Q}_1 =2 and for every n \geq 2 it holds that

    \[\mathcal{Q}_n = 2\mathcal{Q}_{n-1} +\mathcal{Q}_{n-2}\]

Prove that

    \[\sum_{n=1}^{\infty} \arctan \frac{2}{\mathcal{Q}_n} \arctan \frac{2}{\mathcal{Q}_{n+1}} = \frac{\pi^2}{32}\]

 


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