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What is the sum of the infinite series; cot^-1(2)+cot^-1(8)+cot^-1(18)+cot^-1(32)+.......infinity ?

What is the sum of the infinite series;
cot^-1(2)+cot^-1(8)+cot^-1(18)+cot^-1(32)+.......infinity ?

Grade:12th pass

1 Answers

Vishal Kumar
26 Points
5 years ago
firstly in such questions try to find pattern between numbers : here it is 2n2  and then try to express it in form of tan-1a – tan-1b
∑ cot-1 ( 2n2 )  = ∑ tan-1 ( 1/2n2 )   { from n=1 to infinity }
tan-1 ( 2/4n2 )  =  tan-1 { (2n+1) – (2n - 1) \1 + (2n - 1)(2n+1)  } …......................1
tan-1a - tan-1b = tan-1 {(a-b)/1 + a*b}
so equation 1 equals to tan-1 (2n+1) – tan-1 (2n-1)
 so now the qauestion is ∑ {tan-1 (2n+1) – tan-1 (2n-1) }   [ n from 1 to infinity]
so now the answer is  ( \pi/2 – \pi/4 ) = \pi/4 

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