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Question #40708

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \sum_{n=1}^{\infty} \frac{\lfloor \frac{n+1}{2} \rfloor}{n(n+1)(n+2)}$ , where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ . Then the value of $S$ is:

Options

A.
$\frac{1}{2} \ln 2$
B.
$\frac{1}{2} (1 - \ln 2)$
C.
$\ln 2 - \frac{1}{4}$
D.
$\frac{1}{4} (1 + \ln 2)$

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

V-N Method (Partial Fractions)|Method of Differences|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' between continuous telescoping methods (V-N method) and discrete step functions.

A common trap is attempting to force a single partial fraction decomposition on the summand, incorrectly assuming $\lfloor \frac{n+1}{2} \rfloor$ behaves smoothly.

The Greatest Integer Function introduces a domain discontinuity that breaks the telescoping chain.

To resolve this, the student must partition the summation into even and odd domains, effectively neutralizing the GIF.

This reveals a hidden double-layer series: one part forms a standard telescoping sum, while the other forms the Taylor series expansion for $\ln 2$ , filtering out students who cannot bridge discrete mathematics with calculus-based series.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	V-N Method (Partial Fractions)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free