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Let z_r = cos(rα/n²) + i sin(rα/n²), for r = 1, 2, 3,..., n. Then the limit as n → ∞ of the product z₁ z₂ z₃... zₙ is

1. cos α + i sin α
2. cos(α/2) − i sin(α/2)
3. e^(iα/2)
4. ∛(e^(iα))

Correct answer: e^(iα/2)

Solution

Product z_1...z_n = exp(i*(a/n^2)*sum r) = exp(i*(a/n^2)*n(n+1)/2) = exp(i*a(n+1)/(2n)) -> exp(ia/2) as n->infinity, i.e. e^(ia/2).

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