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Find the value of this infinte summation about pell numbers

Source: 2019 Jozsef Wildt International Math Competition-W. 1

April 29, 2020
pell equationnumber theorysplit method

Problem Statement

The Pell numbers PnP_n satisfy P0=0P_0 = 0, P1=1P_1 = 1, and Pn=2Pn−1+Pn−2P_n=2P_{n-1}+P_{n-2} for n≥2n\geq 2. Find ∑n=1∞(tan⁡−11P2n+tan⁡−11P2n+2)tan⁡−12P2n+1\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}
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