Prove that `tan^(- 1)(1/3)+tan^(- 1)(1/7)+tan^(- 1)(1/13)+..........+tan^-1 (1/(n^2+n+1))+......oo =pi/4`A. `(pi)/(2)`B. `(pi)/(4)`C. `(2pi)/(3)`D. 0

Prove that `tan^(- 1)(1/3)+tan^(- 1)(1/7)+tan^(- 1)(1/13)+..........+t

1.	Prove that `tan^(- 1)(1/3)+tan^(- 1)(1/7)+tan^(- 1)(1/13)+..........+tan^-1 (1/(n^2+n+1))+......oo =pi/4`A. `(pi)/(2)`B. `(pi)/(4)`C. `(2pi)/(3)`D. 0
Answer» We have `tan^(-1)1/3+tan^(-1)1/7+tan^(-1)1/13+..+tan^(-1)(1)/(n^(2)+n_+1)+…to infty` `=underset(nrarrinfty)lim underset(r=1)overset(n)Sigma tan^(-1){(1)/(1+r(r+1))}` `=underset(nrarrinfty)lim underset(r=1)oveset(n)Sigma tan^(-1){(r+1)-r)/(1+r(r+1))` `=underset(nrarrinfty)lim (tan^(-1)(n+1)-tan^(-1)1)` `=tan^(-1)infty-tan^(-1)=(pi)/(12)-(pi)/(4)-(pi)/(4)`

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