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Question #40722

Difficulty: 8/10 | 🟢 Free

The Question

Let $r$ be a root of the equation $3x^2 - 8x - 3 = 0$ . If the infinite series $L = \sum_{n=1}^{\infty} \frac{1}{n} \left( \frac{r-1}{r+1} \right)^n$ converges to a finite real number, then the value of $e^{-L}$ is

Options

A.
$\frac{1}{2}$
B.
3
C.
2
D.
$\frac{1}{3}$

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

Logarithmic Series|Quadratic Roots in Sequences|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the domain of convergence for logarithmic series.

The common trap is to algebraically solve the quadratic to find the roots (3 and -1/3), substitute them into the series term $y = \frac{r-1}{r+1}$ , and blindly apply the formula $\sum \frac{y^n}{n} = -\ln(1-y)$ without checking if $y$ lies in the interval of convergence $[-1, 1)$ .

Using the extraneous root $r = -1/3$ yields $y = -2$ , which makes the series diverge, but the blind application of the formula gives a false real value ( $-\ln 3$ ), leading directly to the trap option 3.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Logarithmic Series
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free