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Question #40469

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_k = k^2 + 2i$ for $k \in \mathbb{N}$ , and define $Z_n = \prod_{k=1}^n z_k$ . Evaluate the sum $\sum_{n=1}^{100} \left[ \frac{4}{\pi} \arg(Z_n) \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\arg(z)$ denotes the principal argument of $z$ taking values in $(-\pi, \pi]$ .

Options

A.
198
B.
199
C.
200
D.
201

Check Your Answer

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

Argument and Principal Value|Infinite Series involving Inverse Tangent|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This problem tests the convergence of a telescoping inverse trigonometric series embedded within a complex argument function.

The 'Lethal Collision' arises when this argument is scaled and passed through a Greatest Integer Function (GIF).

Students often incorrectly assume the GIF evaluates uniformly across the series or fail to check the exact boundary conditions (like $n=1$ and $n=2$ ).

Furthermore, they must recognize that the sum of principal arguments does not exceed $\pi$ , avoiding modulo $2\pi$ phase wraps.

The trap lies in missing the discrete jump exactly at $n=2$ and incorrectly evaluating the asymptotic bound, leading to off-by-one errors in the final summation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Argument and Principal Value
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free