Question

It is given that $ z_{r}=cos\left(\pi/2'\right)+isin\left(\pi/2'\right) $ , The value of the product $ z_{1} \cdot z_{2} \cdot z_{3} \cdot z_{4}. \ldots\ldots\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) + i\, sin\left(\frac{\pi}{2^{r}}\right) $
$ \therefore z_{1} \cdot z_{2} \cdot z_{3}\cdot .....\infty $
$=\left[cos\left(\frac{\pi}{2}\right)+i\,sin \left(\frac{\pi}{2}\right)\right]\left[cos \frac{\pi}{4} + i\, sin \frac{\pi}{4}\right]$
$\times\left[cos \frac{\pi}{4} + i \,sin \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 + i\, sin \,\pi = -1 \left(\because sin \,\pi =0\right)$