Question

Find the value of the expression:
$(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})(\cos \frac{\pi }{2^2}+i\sin \frac{\pi }{2^2})\dots \text{to}\infty$

Answer 1

$(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})(\cos \frac{\pi }{2^2}+i\sin \frac{\pi }{2^2})..........................\infty$

$=e^{i\pi /2}\cdot e^{i\pi /2^2}\cdot e^{i\pi /2^3}+.....\infty$

$=e^{i(\pi /2+\pi /2^2+.....)}$

$\left(\frac{a}{|r-1|}\right)$ sum of $\infty$ terms G.P$=\frac{\pi /2}{1-1/2}=\pi$

$=e^{i\pi }=\cos \pi +i\sin \pi$

$=-1$.