The sum of the infinite series `cot^(-1)(7/4)+cot^(-1)((19)/4) +cot^(-1)((39)/4)....oo`

Explanation

Here, general term `T_r` can be wriiten as,

`T_r = cot^-1((4r^2+3)/(4)) = cot^-1(r^2+3/4)`

`=>T_r = tan^-1(1/(r^2+3/4)) = tan^-1(((r+1/2)-(r-1/2))/(1+r^2-1/4))`

`= tan^-1(((r+1/2)-(r-1/2))/(1+(r+1/2)(r-1/2)))`

We know, `tan^-1((x-y)/(1+xy)) = tan^-1x+tan^-1y`

`:. T_r = tan^-1(r+1/2) - tan^-1(r-1/2)`
`:. T_1 = tan^-1 (3/2)-tan^-1 (1/2)`

`T_2 = tan^-1 (5/2)-tan^-1 (3/2)`

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