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Question #41260

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{k=1}^n k$ . Define the function $f(x)$ for $x \ge 1$ as: $f(x) = \sum_{n=2}^{\infty} \tan^{-1}\left( \frac{(-1)^n n^2 x}{1 - S_n S_{n-1} x^2} \right)$ Evaluate the sum of the infinite geometric series $\sum_{k=0}^{\infty} (\sin(f(x)))^k$ at $x = \sqrt{3}$ .

Options

A.
$\frac{2}{3}$
B.
2
C.
$4 + 2\sqrt{3}$
D.
$4 - 2\sqrt{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

The Sum-Difference Formula Constraints|Sum of Powers of Natural Numbers|Infinite GP Convergence

Why This Question Is Tricky

Common Mistakes

This problem tests the lethal collision between telescopic series and the principal domain constraints of the inverse tangent function.

By defining the terms using the sum of natural numbers $S_n$ , the argument forces the product $x_n x_{n-1} < -1$ .

Students who blindly apply the identity $\tan^{-1}(A) - \tan^{-1}(B)$ without the $-\pi$ phase shift correction will obtain oscillating, non-convergent partial sums or an incorrect limit.

The inclusion of an infinite geometric progression evaluates their final converged phase angle, strictly bounding the correct path and punishing naive limit evaluations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Sum-Difference Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free