The value of sum `sum_(n=1)^(oo)cot^(-1)(((n^(2)+2n)(n^(2)+2n+1)+1)/(2n+2))` is equal to

Options

A. `cos^(-1)((1)/(sqrt(5)))`
B. `sec^(-1)((sqrt(5))/(2))`
C. `sin^(-1)((1)/(sqrt(5)))`
D. `cot^(-1)(1)`

Answer

C

Explanation

`T_(n)tan^(-1)((2n+2)/(1+(n^(2)+2n)(n^(2)+2n+1)))`
`=tan^(-1)((2n+2)/(1+n(n+2)(n+1)(n+1)))`
`=(tan^(-1)(n+1)(n+2)-tan^(-1)n(n+1))`
`therefore S_(n)sum_(n=1)^(n)T_(n)=(tan^(-1)(n+1)(n+2)-tan^(-1)2)`
So, `lim_(n rarr oo)S_(n)=((pi)/(2)-tan^(-1)2)=cot^(-1)2=sin^(-1)((1)/(sqrt(5)))`