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Question #41331

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n(x) = \sum_{k=1}^n \frac{x(1 - k(k-1)x^2)}{(1+(k-1)^2 x^2)(1+k^2 x^2)}$ .

Since $S_n(0) = 0$ and $\lim_{x \to \infty} S_n(x) = 0$ , by Rolle's Theorem, $S_n(x)$ has at least one critical point in $(0, \infty)$ . Let $R_n$ be the unique positive root of $S_n'(x) = 0$ .

Evaluate the limit: $\lim_{n \to \infty} n \int_0^{R_n} S_n(x) dx$

Options

A.
$\ln 2$
B.
$\frac{\ln 2}{2}$
C.
$\frac{1}{2}$
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Rolle's Theorem Root Existence|V-N Method (Partial Fractions)|Sum to n terms vs. General Term

Why This Question Is Tricky

Common Mistakes

This question tests the V-N method (telescoping sums) and the application of Rolle's Theorem to a sequence of functions.

The primary trap lies in the temptation to take the limit $n \to \infty$ of $S_n(x)$ before differentiating.

Since $\lim_{n \to \infty} S_n(x) = 0$ for all $x$ , students might erroneously conclude the limit function's derivative is $0$ everywhere, failing to find $R_n$ .

This highlights the danger of assuming uniform convergence and swapping limits with derivatives.

By forcing the evaluation of $R_n$ for a finite $n$ first, the problem filters out students who do not respect the order of operations in sequences of functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Rolle's Theorem Root Existence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free