Will you covert it to tan ⁡ − 1 \tan^{-1} ?

Geometry Level 4

∑ ω = 1 ∞ sin ⁡ − 1 [ 2 ω + 1 ω ( ω + 1 ) ( ω 2 + 2 ω + ω 2 − 1 ) ] = ? \sum_{\omega=1}^{\infty} \sin^{-1} \left [\dfrac{2\omega+1}{\omega(\omega+1)(\sqrt{\omega^{2}+2\omega}+\sqrt{\omega^{2}-1})} \right]= \, ?

The series is divergent π 4 \frac{\pi}{4} π 6 \frac{\pi}{6} 3 π 4 \frac{3\pi}{4} π 2 \frac{\pi}{2}

Shivam Jadhav by Shivam Jadhav , 22, India

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1 solution

First rationalise denominator.Then substitute w=sec(y) and (w+1)=sec(x).It can be seen that after the substitution expression inside summation becomes (x-y)=arcsec(w+1)-arcsec(w).Now,the summation has become a telescopic series.The summation then becomes equal to pi/2.(Note that arcsec(x) means sec inverse of x)

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