The value oftan−$$1$$⁡$$13+$$$$tan$$−$$1$$⁡$$17+$$$$tan$$−$$1$$⁡$$113+$$…$$+$$$$tan$$−$$1$$⁡$$1n2+n+1+$$…to ∞, is

Question

The value oftan−$$1$$⁡$$13+$$$$tan$$−$$1$$⁡$$17+$$$$tan$$−$$1$$⁡$$113+$$…$$+$$$$tan$$−$$1$$⁡$$1n2+n+1+$$…to ∞, is

Infinity Learn · Accepted Answer

We have,$$tan$$−$$1$$⁡$$13+$$$$tan$$−$$1$$⁡$$17+$$$$tan$$−$$1$$⁡$$113+$$…$$+$$$$tan$$−$$1$$⁡$$1n2+n+1+$$…..to⁡∞$$=limn$$→∞ ∑$$r=1n$$ $$tan$$−$$1$$⁡$$1r2+r+1=limn$$→∞ ∑$$r=1n$$ $$tan$$−$$1$$⁡$$11+r(r+1)=limn$$→∞ ∑$$r=1n$$ $$tan$$−$$1$$⁡(r+1)−$$r1+r(r+1)=limn$$→∞ ∑$$r=1n$$ $$tan$$−$$1$$⁡(r+1)−$$tan$$−$$1$$⁡$$r=limn$$→∞ $$tan$$−$$1$$⁡(n+1)−$$tan$$−$$1$$⁡$$1=$$$$tan$$−$$1$$⁡∞−$$tan$$−$$1$$⁡$$1=$$π$$2$$−π$$4=$$π$$4$$

The value of
$\tan^{- 1} \frac{1}{3} + \tan^{- 1} \frac{1}{7} + \tan^{- 1} \frac{1}{13} + \dots + \tan^{- 1} \frac{1}{n^{2} + n + 1} + \dots$ to $\infty$ , is

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The value oftan−1⁡13+tan−1⁡17+tan−1⁡113+…+tan−1⁡1n2+n+1+…to ∞, is

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The value of
$\tan^{- 1} \frac{1}{3} + \tan^{- 1} \frac{1}{7} + \tan^{- 1} \frac{1}{13} + \dots + \tan^{- 1} \frac{1}{n^{2} + n + 1} + \dots$ to $\infty$ , is