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Find sum of infinite terms of series cot^-1(2^2+1/2)+cot^-1(2^3+1/4)+cot^-1(2^4+1/8)............

Find sum of infinite terms of series cot^-1(2^2+1/2)+cot^-1(2^3+1/4)+cot^-1(2^4+1/8)............

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
5 years ago
We know that arccotx=arctan (1/x) for x greater than zero. So we convert all terms into arctan. Now the nth term is
arctan (2^n/1+2^n*2^n+1)
= arctan (2^n+1-2^n/1+2^n*2^n+1)
= arctan 2^n+1-arctan2^n
But the above sum telescopes.
So sum is arctan infinite-arctan2
=pi/2-arctan2
= arccot2

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