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Question #41448

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{n \to \infty} \frac{1}{n^2} \sum_{r=n}^{2n-1} r \frac{\binom{4n}{r}}{\binom{4n}{r+1}}$

Options

A.
$16 \ln\left(\frac{3}{2}\right) - \frac{11}{2}$
B.
$16 \ln\left(\frac{4}{3}\right) - \frac{9}{2}$
C.
$\frac{5}{2}$
D.
$4 \ln\left(\frac{3}{2}\right) - 1$

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

Integral as the Limit of a Sum|Sigma Notation Properties|General Term Formulation

Why This Question Is Tricky

Common Mistakes

This question tests the conversion of a limit of a sum into a definite integral using Riemann sums, specifically targeting the 'Sigma Notation' and 'General Term Formulation' traps.

Students must first correctly simplify the ratio of binomial coefficients.

A common trap is blindly applying the standard integration limits from 0 to 1 without analyzing the summation indices ( $r=n$ to $2n-1$ ), which correctly map to the interval $[1, 2]$ .

Furthermore, failing to properly normalize the $1/n$ factor or ignoring the $+1/n$ term's vanishing behavior as $n \to \infty$ will lead to incorrect integral setups.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integral as the Limit of a Sum
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free