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Find the sum of the infinite series $$\cos^{-1}(2.1^2)+\cos^{-1}(2.2^2)+\cos^{-1}(2.3^2)+\cos^{-1}(2.4^2)+......$$

My Attempt \begin{align} \cot^{-1}x&=\tan^{-1}\frac{1}{x}\text{, }x>0\\ \cot^{-1}(2.1^2)&+\cot^{-1}(2.2^2)+\cot^{-1}(2.3^2)+\cot^{-1}(2.4^2)+.....=\\ &=\tan^{-1}\frac{1}{2.1^2}+\tan^{-1}\frac{1}{2.2^2}+\tan^{-1}\frac{1}{2.3^2}+\tan^{-1}\frac{1}{2.4^2}+.....\\ &=\tan^{-1}\frac{}{}+\tan^{-1}\frac{}{}+\tan^{-1}\frac{}{}+\tan^{-1}\frac{}{}+..... \end{align} The solution given in my reference is $\dfrac{\pi}{4}$, but I am stuck as in my attempt not able to identify any common property among the terms to simplify the series.

Sooraj S
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    $$\dfrac1{2n^2}=\dfrac{2n+1-(2n-1)}{1+(2n-1)(2n+1)}$$ – lab bhattacharjee Dec 18 '18 at 09:31
  • See also : https://math.stackexchange.com/questions/193001/explicitly-finding-the-sum-of-arctan1-n2n1 https://math.stackexchange.com/questions/415512/is-s-sum-r-1-infty-tan-1-frac2r2r2r4-finite https://math.stackexchange.com/questions/2440420/infinite-sum-of-cot-1-n2-3-4 – lab bhattacharjee Dec 18 '18 at 09:34

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