For any positive integer n, let Sn:(0,∞)→R be defined by Snx=∑k=1ncot-11+k(k+1)x2x where for any x∈R,cot-1(x)∈(0,π) and tan-1(x)∈-π2,π2. Then which of the following statements is (are) TRUE ?

(B) limn→∞cotSn(x)=x, for all x>0

(A) S10x=π2-tan-11+11x210x, for all x>0

(B) limn→∞cotSn(x)=x, for all x>0

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

(D) tanSn(x)≤12, for all n≥1 and x>0

JEE Advanced · Mathematics · 21. ITF

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 ?

A $S_{10} (x) = \frac{π}{2} - \tan^{- 1} (\frac{1 + 11 x^{2}}{10 x})$ , for all $x > 0$
B $\lim_{n \to \infty} \cot (S_{n} (x)) = x$ , for all $x > 0$
C The equation $S_{3} (x) = \frac{π}{4}$ has a root in $(0, \infty)$
D $\tan (S_{n} (x)) \leq \frac{1}{2}$ , for all $n \geq 1$ and $x > 0$

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

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

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

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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For any positive integer n, let Sn:(0,∞)→R be defined bySnx=∑k=1ncot-11+k(k+1)x2xwhere for any x∈R,cot-1(x)∈(0,π) and tan-1(x)∈-π2,π2. Then which of the following statements is (are) TRUE ?

Step-by-step Solution