Question

If $x_n=\cos \left(\frac{\pi }{4^n}\right)+i\sin \left(\frac{\pi }{4^n}\right)$, then $x_1\cdot x_2\cdot x_3......\infty$ is?

• A. $\frac{1+i\sqrt{3}}{2}$
• B. $\frac{-1+i\sqrt{3}}{2}$
• C. $\frac{1-i\sqrt{3}}{2}$
• D. $\frac{-1-i\sqrt{3}}{2}$

Answer 1

Answer: (A)

$\frac{1+i\sqrt{3}}{2}$

Given, $x_n=\cos \left(\frac{\pi }{4^n}\right)+i\sin \left(\frac{\pi }{4^n}\right)=e^{i\frac{\pi }{4^n}}$

$\therefore x_1\cdot x_2\cdot x_3\dots \infty =e^{i(\frac{\pi }{4}+\frac{\pi }{4^2}\dots \infty )}$

$=\cos (\frac{\pi }{4}+\frac{\pi }{4^2}+\frac{\pi }{4^3}\dots \infty )+i\sin (\frac{\pi }{4}+\frac{\pi }{4^2}+\frac{\pi }{4^3}\dots \infty )$

$=cos\left(\frac{\pi /4}{1-1/4}\right)+i\sin \left(\frac{\pi /4}{1-1/4}\right)=\cos \left(\pi /3\right)+i\sin \left(\pi /3\right)=\frac{1+i\sqrt{3}}{2}$