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If $z_r=cos(\large\frac{\pi}{3^r})$$+isin(\large\frac{\pi}{3^r})$, where $r=1,2,3...........then \:\:z_1.z_2.z_3.....................\infty=?$

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  • $z_1.z_2=cos(\alpha+\beta)+isin(\alpha+\beta)$ where $z_1=cis\alpha,\:z_2=cis\beta$
  • Sum of infinite G.P $(a+ar+ar^2+.......)=\large\frac{a}{1-r}$
Given: $z_r=cos(\large\frac{\pi}{3^r})$$+isin(\large\frac{\pi}{3^r})$
$z_1.z_2.z_3.......=(cos\large\frac{\pi}{3}$$+isin\large\frac{\pi}{3}).\:(cos\large\frac{\pi}{3^2}$$+isin\large\frac{\pi}{3^2}).............$
$=cos(\large\frac{\pi}{3}+\frac{\pi}{3^2}+\frac{\pi}{3^3}+.........)$$+isin(\large\frac{\pi}{3}+\frac{\pi}{3^2}+.............)$
$\large\frac{\pi}{3}+\frac{\pi}{3^2}+\frac{\pi}{3^3}+.............=\large\frac{\frac{\pi}{3}}{1-\frac{1}{3}}$
$=\large\frac{\pi}{2}$
$\therefore\:z_1.z_2.z_3...........=cos\large\frac{\pi}{2}$$+isin\large\frac{\pi}{2}$$=i$
answered Aug 1, 2013 by rvidyagovindarajan_1
 

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