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Question #41908

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{k=1}^n (-1)^{k-1} \left[ k^2 + (k+1)^2 - k^3 \sin\left(\frac{1}{k}\right) - (k+1)^3 \sin\left(\frac{1}{k+1}\right) \right]$ . Evaluate the limit: $\lim_{n \to \infty} n^2 \left| S_n - 1 + \sin(1) + \frac{(-1)^n}{6} \right|$

Options

A.
$\frac{1}{120}$
B.
$\frac{1}{60}$
C.
$\frac{1}{24}$
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Modulus Function Breakdown|Trigonometric Standard Limits|Sum to n terms vs. General Term

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify a hidden telescoping series with alternating signs and handle limits of sequences that conditionally oscillate.

The primary trap lies in the parity-dependent term $(-1)^n$ .

Students who attempt to evaluate the limit of $S_n$ directly will falsely conclude it does not exist due to oscillation.

The modulus acts as a parity-neutralizer, but only after correctly grouping the terms.

Furthermore, it forces a higher-order Taylor expansion (up to $x^5$ ) to determine the exact rate of decay, penalizing those who only use the standard limit $(x-\sin x)/x^3 \to 1/6$ and incorrectly assume the final answer is 0.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Modulus Function Breakdown
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free