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Let xₙ = cos(pi / 2ⁿ) + i*sin(pi / 2ⁿ). If the infinite product x₁ * x₂ * x₃ *... equals lambda, then |lambda| is:

1. (A) 0
2. (B) 1
3. (C) 1/2
4. (D) 2

Correct answer: (B) 1

Solution

xₙ = e^(i*pi/2ⁿ). The infinite product = e^(i*pi*(1/2 + 1/4 + 1/8 +...)) = e^(i*pi*1) = e^(i*pi) = cos(pi) + i*sin(pi) = -1. So lambda = -1 and |lambda| = 1.

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