Question

If \( z_r = \cos \frac{r\alpha}{n^2} + i \sin \frac{r\alpha}{n^2} \), where \( r = 1, 2, 3, ..., n \), then the value of \( \lim_{n \to \infty} z_1 z_2 z_3 ... z_n \) is:

• A. 0
• B. \( e^{\frac{i \alpha}{2}} \)
• C. \( e^{\frac{i \alpha}{2}} \)
• D. \( e^{i \alpha} \)

Hint:

Euler's identity, \( e^{i\theta} = \cos \theta + i \sin \theta \), is useful when working with products of complex numbers in trigonometric form.