For any positive integer n, Let Sn:(0,∞)→RR defined by Sn(x)=∑k=1n cot−1⁡1+k(k+1)x2xwhere x∈R, cot−1⁡x∈(0,π),tan−1∈−π2,π2then which of the following statements is (are) the TRUE?

S 11 ( x ) = π 2 − tan − 1 ⁡ 1 + 12 x 2 11 x , tan ⁡ S n ( x ) ≤ 1 2 ∀ n ≥ 1 , x > 0 , Lt n → ∞ cot ⁡ S n ( x ) = x for all x > 0

For any positive integer n, Let Sn:(0,∞)→RR defined by Sn(x)=∑k=1n cot−1⁡1+k(k+1)x2xwhere x∈R, cot−1⁡x∈(0,π),tan−1∈−π2,π2then which of the following statements is (are) the TRUE?

A) S n ( x ) = ∑ k = 1 n tan − 1 ⁡ ( k + 1 ) x − kx 1 + ( k + 1 ) ( k ) x 2 = tan − 1 ⁡ ( nx + x ) − tan − 1 ⁡ x = tan − 1 ⁡ nx 1 + ( n + 1 ) x 2 S 11 x = tan − 1 ⁡ 11 x 1 + 12 x 2 = π 2 − cot − 1 ⁡ 11 x 1 + 12 x 2 = π 2 − tan − 1 ⁡ 1 + 12 x 2 112 B) Lt n → ∞ ⁡ 1 + ( n + 1 ) x 2 nx = Lt n → ∞ ⁡ 1 n + 1 + 1 n x 2 x = x 2 x = x C) tan − 1 ⁡ 3 x 1 + 4 x 2 = π 4 → 3 x = 1 + 4 x 2 ⇒ Δ < 0 , norealroot D) Let nx 1 + ( n + 1 ) x 2 ≤ 1 2 is true ⇒ 2 nx ≤ ( n + 1 ) x 2 + 1 ( n + 1 ) x 2 − 2 nx + 1 ≥ 0 Δ = 4 n 2 − 4 ( n + 1 ) , ( n ∈ N ) Δ < 0 for n = 1 , Δ > 0 , for n ≥ 2 for some x > 0

S 11 ( x ) = π 2 − tan − 1 ⁡ 1 + 12 x 2 11 x

tan ⁡ S n ( x ) ≤ 1 2 ∀ n ≥ 1 , x > 0

Lt n → ∞ cot ⁡ S n ( x ) = x for all x > 0

The equation S 3 ( x ) = π 4 has no real roots in ( 0 , ∞ )

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For any positive integer n, $Let S_{n} : (0, \infty) \to R$ R defined by $S_{n} (x) = \sum_{k = 1}^{n} \cot^{- 1} (\frac{1 + k (k + 1) x^{2}}{x})$ where $x \in R, \cot^{- 1} x \in (0, π), \tan^{- 1} \in (\frac{- π}{2}, \frac{π}{2})$ then which of the following statements is (are) the TRUE?

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$S_{11} (x) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 12 x^{2}}{11 x})$

$\tan (S_{n} (x)) \leq \frac{1}{2} \forall n \geq 1, x > 0$

$\underset{n \to \infty}{Lt} \cot (S_{n} (x)) = x for all x > 0$

The equation $S_{3} (x) = \frac{π}{4}$ has no real roots in $(0, \infty)$

answer is A, B, C.

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Detailed Solution

A) $S_{n} (x) = \sum_{k = 1}^{n} \tan^{- 1} (\frac{(k + 1) x - kx}{1 + (k + 1) (k) x^{2}}) = \tan^{- 1} (nx + x) - \tan^{- 1} x$
$\begin{array}{l} = \tan^{- 1} (\frac{nx}{1 + (n + 1) x^{2}}) \\ S_{11} x = \tan^{- 1} (\frac{11 x}{1 + 12 x^{2}}) = \frac{π}{2} - \cot^{- 1} (\frac{11 x}{1 + 12 x^{2}}) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 12 x^{2}}{112}) \end{array}$
B) ${Lt}_{n \to \infty} \frac{1 + (n + 1) x^{2}}{nx} = {Lt}_{n \to \infty} \frac{\frac{1}{n} + (1 + \frac{1}{n}) x^{2}}{x} = \frac{x^{2}}{x} = x$
C) $\tan^{- 1} (\frac{3 x}{1 + 4 x^{2}}) = \frac{π}{4} \to 3 x = 1 + 4 x^{2} \Rightarrow Δ < 0, norealroot$
D) Let $\frac{nx}{1 + (n + 1) x^{2}} \leq \frac{1}{2} is true \Rightarrow 2 nx \leq (n + 1) x^{2} + 1$
$\begin{array}{l} (n + 1) x^{2} - 2 nx + 1 \geq 0 \\ Δ = 4 n^{2} - 4 (n + 1), (n \in N) \end{array}$

$Δ < 0 for n = 1, Δ > 0, for n \geq 2 for some x > 0$