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Question #40922

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{n \to \infty} \sum_{k=1}^{2n} \frac{n^3}{k^4} \left( \ln\left(1 + \frac{k^2}{n^2}\right) - \frac{k^2}{n^2} \right)$

Options

A.
$\frac{4 - \ln 5 - 16 \arctan 2}{24}$
B.
-1
C.
$\frac{4 - \ln 5 + 16 \arctan 2}{24}$
D.
$\frac{4 - \ln 5}{24}$

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

Definition of Limit|Sigma Notation Properties|Logarithmic Series|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the conversion of a limit of a sum into a definite integral, specifically targeting the misuse of Taylor series expansions.

A common trap is assuming $\ln(1+x^2) \approx x^2 - x^4/2$ because $n \to \infty$ , leading to the integral of $-1/2$ and an incorrect answer of $-1$ .

Students fail to recognize that the argument $x = k/n$ ranges up to $2$ , which lies outside the radius of convergence ( $|x| \le 1$ ) for the Maclaurin series of $\ln(1+x^2)$ .

Thus, term-by-term integration of the truncated series is mathematically invalid, forcing the use of integration by parts on the exact function.

Quick Facts About This Question

Question Information

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