For any positive integer n , let S n : ( 0 , ) be defined by S n ( x ) = k = 1 n cot - 1 ( 1 + k ( k + 1 ) x 2 x ) , where for any x , cot - 1 x ( 0 , ) and tan - 1 ( x ) ( - 2 , 2 ) . Then which of the following statements is (are) TRUE?

Question

For any positive integer $n$ , let $S_{n} : (0, \infty) \to ℝ$ be defined by $S_{n} (x) = \sum_{k = 1}^{n} \cot^{- 1} (\frac{1 + k (k + 1) x^{2}}{x}),$ where for any $x \in ℝ$ ,
$\cot^{- 1} x \in (0, π)$ and $\tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2}) .$ Then which of the following statements is (are) TRUE? [2021]

Smart Achievers · Accepted Answer

(1, 2)

$Given that S_{n} (x) = \sum_{k = 1}^{n} \cot^{- 1} (\frac{1 + k (k + 1) x^{2}}{x})$

$= \sum_{k = 1}^{n} \tan^{- 1} (\frac{x}{1 + k x (k x + x)})$

$\tan^{- 1} (x) \in (- \frac{π}{2}, \frac{π}{2}) .$

$\Rightarrow S_{n} (x) = \tan^{- 1} (n x + x) - \tan^{- 1} x$ $= \tan^{- 1} (\frac{n x}{1 + (n + 1) x^{2}})$

$(1) S_{10} (x) = \tan^{- 1} (\frac{10 x}{1 + 11 x^{2}}) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 11 x^{2}}{10 x}), (x > 0)$

$Option (1) is correct$

$(2) \lim_{n \to \infty} \cot (S_{n} (x)) = \lim_{n \to \infty} \cot [\cot^{- 1} (\frac{1 + (n + 1) x^{2}}{n x})]$

$= \lim_{n \to \infty} \frac{\frac{1}{n} + (1 + \frac{1}{n}) x^{2}}{x} = x, (x > 0)$

$Option (2) is correct$

$(3) S_{3} (x) = \tan^{- 1} (\frac{3 x}{1 + 4 x^{2}}) = \frac{π}{4}$

$\Rightarrow 4 x^{2} - 3 x + 1 = 0 \Rightarrow x \notin ℝ$ $[∵ D is negative]$

$Option (3) is incorrect$

$(4) For x = 1,$

$\tan (S_{n} (x)) = \frac{n}{n + 2} \geq \frac{1}{2}, for n \geq 3$

$Option (4) is incorrect$

Solution

1 $S_{10} (x) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 11 x^{2}}{10 x}), for all x > 0$

2 $\lim_{n \to \infty} \cot (S_{n} (x)) = x, for all x > 0$

3 The equation $S_{3} (x) = \frac{π}{4}$ has a root in $(0, \infty)$

4 $\tan (S_{n} (x)) \leq \frac{1}{2}, for all n \geq 1 and x > 0$

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Solution

Q. For any positive integer n, let Sn:(0,∞)→ℝ be defined by Sn(x)=∑k=1ncot-1(1+k(k+1)x2x), where for any x∈ℝ, cot-1x∈(0,π) and tan-1(x)∈(-π2,π2). Then which of the following statements is (are) TRUE [2021]

1 S10(x)=π2-tan-1(1+11x210x), for all x>0

2 limn→∞cot(Sn(x))=x, for all x>0

3 The equation S3(x)=π4 has a root in (0,∞)

4 tan(Sn(x))≤12, for all n≥1 and x>0

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1 $S_{10} (x) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 11 x^{2}}{10 x}), for all x > 0$

2 $\lim_{n \to \infty} \cot (S_{n} (x)) = x, for all x > 0$

3 The equation $S_{3} (x) = \frac{π}{4}$ has a root in $(0, \infty)$

4 $\tan (S_{n} (x)) \leq \frac{1}{2}, for all n \geq 1 and x > 0$