The value of sum `sum_(n=1)^(oo)cot^(-1)(((n^(2)+2n)(n^(2)+2n+1)+1)/(2n+2))` is equal toA. `cos^(-1)((1)/(sqrt(5)))`B. `sec^(-1)((sqrt(5))/(2))`C. `sin^(-1)((1)/(sqrt(5)))`D. `cot^(-1)(1)`

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

1.	The value of sum `sum_(n=1)^(oo)cot^(-1)(((n^(2)+2n)(n^(2)+2n+1)+1)/(2n+2))` is equal toA. `cos^(-1)((1)/(sqrt(5)))`B. `sec^(-1)((sqrt(5))/(2))`C. `sin^(-1)((1)/(sqrt(5)))`D. `cot^(-1)(1)`
Answer» Correct Answer - C `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)))`

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