Question

It is given that $z_r=cos\left(\pi /2^{\prime }\right)+isin\left(\pi /2^{\prime }\right)$ , The value of the product $z_1\cdot z_2\cdot z_3\cdot z_4.\dots \dots \infty$ is

• A. $0$
• B. $1$
• C. $-1$
• D. $1+i$

Answer 1

Answer: (C)

We have, $z_r=cos\left(\frac{\pi }{2^r}\right)+isin\left(\frac{\pi }{2^r}\right)$
$\therefore z_1\cdot z_2\cdot z_3\cdot .....\infty$
$=\left[cos\left(\frac{\pi }{2}\right)+isin\left(\frac{\pi }{2}\right)\right]\left[cos\frac{\pi }{4}+isin\frac{\pi }{4}\right]$
$\times \left[cos\frac{\pi }{4}+isin\frac{\pi }{8}\right]\times .....\infty$
$=e^{i\pi /2}\cdot e^{i\pi /4}\cdot e^{i\pi /8}.....\infty$
$=e^{i\pi \left[\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+.....\infty \right]}=e^{i\pi \left[\frac{12}{1-12}\right]}=e^{i\pi }$
$=cos\pi +isin\pi =-1\left(\because sin\pi =0\right)$