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For any positive integer n . let Sn:(0,o...

For any positive integer n . let `S_n:(0,oo) to R` be defined by
`S_n(x)=sum_(k=1)^n Cot^-1((1+k(k+1)x^2)/x)`
where for any `x in R , cot^-1x in (0,pi) and tan^-1(x) in (-pi/2, pi/2)`.Then which of the following statement is(are TRUE ?

A

`S_10(x)=pi/2-tan^-1((1+11x^2)/(10x))` for all `x gt 0`

B

`lim_(n to oo) cot(S_n(x)=x` for all `x gt 0`

C

The equation `S_3(x)=pi/4` has a root in `(0,oo)`

D

`tan(S_n(x)) le 1/2` for all `n ge1 and x ge 0`

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