Find the sum of the series :`tan^- 1(1/3)+tan^- 1(2/9)+....+tan^- 1((2^(n-1))/(1+2^(2n-1)))+...... oo`

Explanation

`T_(n)=tan^(-1)((2^(n-1))/(1+2^(2n-1)))=tan^(-1)((2^(n-1))/(1+2^(n)*2^(n-1)))`
`=tan^(-1)((2^(n)-2^(n-1))/(1+2^(n)*2^(n-1)))=tan^(-1)2^(n)-tan^(-1)2^(n-1)`
`S_(n)=T_(1)+T_(2)+"......."+T_(n)`
`=(tan^(-1)2^(1)-tan^(-1)2^(0))+(tan^(-1)2^(2)-tan^(-1)2^(1))+"........"+(tan^(-1)2^(n)-tan^(-1)2^(n-1))`
`=(tan^(-1)2^(n)-tan^(-1)1)`
`S_(n)=tan^(-1)2^(n)-(pi)/(4)`
`S=lim_(n to oo)S_(n)=lim_(n to oo)(tan^(-1)2^(n)-(pi)/(4))=(pi)/(2)-(pi)/(4)=(pi)/(4)`.