The sum of the series $$cot$$−$$1$$⁡$$2+$$$$cot$$−$$1$$⁡$$8+$$$$cot$$−$$1$$⁡$$18+$$$$cot$$−$$1$$⁡$$32+$$… is

Question

The sum of the series $$cot$$−$$1$$⁡$$2+$$$$cot$$−$$1$$⁡$$8+$$$$cot$$−$$1$$⁡$$18+$$$$cot$$−$$1$$⁡$$32+$$… is

Infinity Learn · Accepted Answer

The given $$series=$$$$cot$$−$$1$$⁡$$2(1)2+cot2$$⁡$$2(2)2+$$$$cot$$−$$1$$⁡$$2(3)2+$$$$cot$$−$$1$$⁡$$2(4)2+$$…∴$$Tn=$$$$cot$$−$$1$$⁡$$2n2=$$$$tan$$−$$1$$⁡$$12n2$$ $$=$$$$tan$$−$$1$$⁡(2n+1)−(2n$$−$$1)1+(2n+1)(2n$$−$$1) $$=$$$$tan$$−$$1$$⁡(2n+1)−$$tan$$−$$1$$⁡(2n$$−$$1)∴$$T1=$$$$tan$$−$$1$$⁡$$3$$−$$tan$$−$$1$$⁡$$1T2=$$$$tan$$−$$1$$⁡$$5$$−$$tan$$−$$1$$⁡$$3T3=$$$$tan$$−$$1$$⁡$$7$$−$$tan$$−$$1$$⁡$$5and$$ so on.∴$$Sn=$$$$tan$$−$$1$$⁡(2n+1)−$$tan$$−$$1$$⁡$$1$$∴limn→∞ $$Sn=$$$$tan$$−$$1$$⁡∞−π$$4=$$π$$2$$−π$$4=$$π$$4$$

The sum of the series $\cot^{- 1} 2 + \cot^{- 1} 8 + \cot^{- 1} 18 + \cot^{- 1} 32 + \dots$ is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

The sum of the series cot−1⁡2+cot−1⁡8+cot−1⁡18+cot−1⁡32+… is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

The sum of the series $\cot^{- 1} 2 + \cot^{- 1} 8 + \cot^{- 1} 18 + \cot^{- 1} 32 + \dots$ is