If , where and , then is equal…

MATHEMATICS · Single correct · IB Diploma Programme · Mathematics: Analysis and Approaches · Number and Algebra · Number and Algebra

Problem

z_r = \cos\left(\frac{r\alpha}{n^2}\right) + i \sin\left(\frac{r\alpha}{n^2}\right)

, where

r = 1, 2, 3, \dots, n

and

i = \sqrt{-1}

, then

\lim_{n \to \infty} z_1 z_2 z_3 \dots z_n

is equal to (1)

e^{i\alpha}

(2)

e^{-i\alpha/2}

(3)

e^{i\alpha/2}

(4)

\sqrt[3]{e^{i\alpha}}