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Question #40931

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_k(n) = \sum_{i=1}^n i^k$ . Evaluate the limit: $\lim_{N \to \infty} \sum_{n=1}^N \frac{2 S_1(n) + 1}{4 S_3(n)}$

Options

A.
$\frac{\pi^2}{3} - 2$
B.
$\frac{\pi^2}{3}$
C.
2
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Algebraic Simplification|The Telescopic Series Trap|Sum of Powers of Natural Numbers

Why This Question Is Tricky

Common Mistakes

This question tests the 'Telescopic Series Trap' combined with the algebraic simplification of sum of powers.

Students often assume that complex rational terms in infinite series are purely telescopic and will completely cancel out.

When decomposing the squared fraction $\frac{1}{n^2(n+1)^2}$ , a common heuristic mistake is to incorrectly partial fraction it as $\frac{1}{n^2} - \frac{1}{(n+1)^2}$ (which sums to 1) or assume the squared terms vanish, leading to incorrect answers like 0 or 2.

The problem forces students to rigorously expand the squared difference, revealing a non-telescopic residue that requires knowledge of the Basel problem ( $\sum 1/n^2 = \pi^2/6$ ) to evaluate correctly.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebraic Simplification
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free