If `sum_(r=1)^(oo) cot^(-1)(((r+1)^(2))/(2))=tan^(-1)x` then `x` is equal to

sum_(i=1)^(oo)tan^(-1)((1)/(2r^(2)))=

sum_(r=1)^(oo)tan^(-1)((2)/(1+(2r+1)(2r-1)))

The value of sum_(r=1)^(oo)cot^(-1)((r^(2))/(2)+(15)/(8)) is equal to

The value of sum_(r=1)^(oo)tan^(-1)((2r-1)/(r^(4)-2r^(3)+r^(2)+1)) is equal to

If the sum of the series sum_(r=1)^(oo)cot^(-1)(2^(r+1)+(1)/(2^(r))) is cot^(-1)(k), then the fundamental period of the function f(x)=sec(sec kx)+sec(cos eckx), is

If y=sum_(r=1)^(x) tan^(-1)((1)/(1+r+r^(2))) , then (dy)/(dx) is equal to