Question

Find the value of the expression
$(cos\frac{\pi }{2}+isin\frac{\pi }{2})(cos\frac{\pi }{2^2}+isin\frac{\pi }{2^2})...$ to $\infty$

• A. 1
• B. -1
• C. i
• D. -i

Answer 1

The correct option is B -1

$(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})(\cos \frac{\pi }{2^2}+i\sin \frac{\pi }{2^2})...$ to $\infty$
$=\cos (\frac{\pi }{2}+\frac{\pi }{2^2}+...)+i\sin (\frac{\pi }{2}+\frac{\pi }{2^2}+...)$
$=\cos \left[\frac{\pi }{2}(1+\frac{1}{2}+\frac{1}{2^2}+...)\right]+i\sin \left[\frac{\pi }{2}(1+\frac{1}{2}+\frac{1}{2^2}+...)\right]$
$=\cos \left[\frac{\pi }{2}\left(\frac{1}{1-\frac{1}{2}}\right)\right]+i\sin \left[\frac{\pi }{2}\left(\frac{1}{1-\frac{1}{2}}\right)\right]$
$=\cos \pi +i\sin \pi =-1$
Ans: B