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Question #67957

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{k=1}^n \frac{\arctan(k)}{k^2}$ . By analyzing the concavity of the function $f(x) = \frac{\arctan x}{x^2}$ and evaluating the integral $I_n = \int_{-n}^{-1} f(x) dx$ , which of the following inequalities is strictly true for all integers $n \ge 2$ ?

Options

A.
$S_n < \frac{3\pi}{8} + \frac{1}{2}\ln 2 - \frac{\arctan n}{n} \left( 1 - \frac{1}{2n} \right) - \frac{1}{2}\ln\left(1 + \frac{1}{n^2}\right)$
B.
$S_n > \frac{3\pi}{8} + \frac{1}{2}\ln 2 - \frac{\arctan n}{n} \left( 1 - \frac{1}{2n} \right) - \frac{1}{2}\ln\left(1 + \frac{1}{n^2}\right)$
C.
$S_n < \frac{3\pi}{8} - \frac{1}{2}\ln 2 - \frac{\arctan n}{n} \left( 1 - \frac{1}{2n} \right) + \frac{1}{2}\ln\left(1 + \frac{1}{n^2}\right)$
D.
$S_n > \frac{3\pi}{8} - \frac{1}{2}\ln 2 - \frac{\arctan n}{n} \left( 1 - \frac{1}{2n} \right) + \frac{1}{2}\ln\left(1 + \frac{1}{n^2}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Infinite Series involving Inverse Tangent|Linearity and Partitioning|Concavity Analysis|Logarithmic Integration Modulus

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the ability to bound an infinite series using definite integrals by analyzing the concavity of the summand function.

The core trap lies in the 'Logarithmic Integration Modulus'.

When integrating the derivative of the inverse trigonometric function via partial fractions, the term $\ln|x|$ emerges.

Because the integration bounds are strictly negative ( $-n$ to $-1$ ), failing to apply the absolute value correctly leads to undefined logarithms or incorrect sign distributions when combining logarithmic terms, which directly maps to the incorrect options provided.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Infinite Series involving Inverse Tangent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free