If Z_r = cos left(frac{pi}{2^r}right) + i sin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right) = e^{i \frac{\pi}{2^r}}$.The product is $P = Z_1 Z_2 Z_3 \ldo

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(D) Given $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right) = e^{i \frac{\pi}{2^r}}$.The product is $P = Z_1 Z_2 Z_3 \ldots = e^{i \frac{\pi}{2^1}} \cdot e^{i \frac{\pi}{2^2}} \cdot e^{i \frac{\pi}{2^3}} \ldots$Using the property of exponents,$P = e^{i \left( \frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + \ldots \right)}$.The exponent is a geometric series with first term $a = \frac{\pi}{2}$ and common ratio $r = \frac{1}{2}$.The sum of the infinite geometric series is $S = \frac{a}{1 - r} = \frac{\pi/2}{1 - 1/2} = \frac{\pi/2}{1/2} = \pi$.Therefore,$P = e^{i \pi} = \cos \pi + i \sin \pi = -1 + i(0) = -1$.

If $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right)$ for $r = 1, 2, 3, \ldots$,then the product $Z_1 Z_2 Z_3 \ldots \infty$ is equal to:

Similar Questions

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^{12}+\beta^{12}=$

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^4+\left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^4=r \operatorname{cis} \theta$,then one of the values of $\sqrt{r \operatorname{cis} \theta}$ is

If $\alpha, \beta$ are the roots of $x^2-2x+4=0$, for $n \in N$, what is the value of $\alpha^n+\beta^n$?

If $z = \sin \theta - i \cos \theta,$ then for any integer $n$

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + ..... + C_nx^n$,then the value of $C_0 - C_2 + C_4 - C_6 + .....$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module