Find the sum `sum_(r =1)^(oo) tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) -2r + 1)))`

Explanation

`S_(n) = underset(r =1)overset(n)sum tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) - 2r + 1)))`
`= underset(r =1)overset(n) tan^(-1) ((2(2r -1))/(4 + r^(2) (r -1)^(2)))`
`= underset(r=1)overset(n)sum tan^(-1). (((2r-1)/(2))/(1+ (r^(2))/(2).((r-1)^(2))/(2)))`
`= underset(r=1)overset(n)sum tan^(-1)(((r^(2))/(2) - ((r-1)^(2))/(2))/(1 + (r^(2))/(2).((r -1)^(2))/(2)))`
`= underset(r =1)overset(n) sum (tan^(-1). (r^(2))/(2) - tan^(-1). ((r -1)^(2))/(2))`
`= tan^(-1).(n^(2))/(2)`
`:. S_(oo) = underset(n rarr oo)("lim") tan^(-1).(n^(2))/(2) = (pi)/(2)`