Solveeit Logo
Login

Question

Question: If x<sub>n</sub> = cos\(\frac{\pi}{2^{n}}\)+ i sin\(\frac{\pi}{2^{n}}\), then\(\prod_{n = 1}^{\infty...

If xn = cosπ2n\frac{\pi}{2^{n}}+ i sinπ2n\frac{\pi}{2^{n}}, then∏n=1∞xn\prod_{n = 1}^{\infty}x_{n} is equal to –

A

– 1

B

1

C

12\frac{1}{\sqrt{2}}

D

i2\frac{i}{\sqrt{2}}

Answer

– 1

Explanation

Solution

Sol. x1 x2 x3 …  = c is π2\frac{\pi}{2} × c isπ22\frac{\pi}{2^{2}}× c isπ23\frac{\pi}{2^{3}}…. =c i

(π2+π22+π23..........∞)\left( \frac{\pi}{2} + \frac{\pi}{2^{2}} + \frac{\pi}{2^{3}}..........\infty \right)

= c is π/21−12\frac{\pi/2}{1 - \frac{1}{2}} = isp = cosp + i sinp = –1