If `x_(n)=cospi/3^(n)+isin(pi)/(3^(n))`, then `x_(1),x_(2),x_(3),………………..x_(infty)` is equal to

Explanation

(i) we have,
`x_(k) cos (pi)/(3^(k)) + isin ( pi)/(3^(k)) = 3^(3^((pi)/k))`
`x_(1)x_(2)x_(3)x_(4)….."to" oo`
`e^(ipi//3).e^(ipi//3^(2)).e^(ipi//3^(3))......"up to" oo`
`e^(i(pi/3+pi/3^(2)+pi/3^(3)+pi/3^(4)..... "to" oo))`
`e^(((pi/3)/(1-1/3)))= e ^(i(pi/2))= cos "" pi/2 = i sin ""pi/2`
= i
(ii) we have, ` ((1 + sin alpha + i cos alpha)/(1 + sin alpha - icos alpha))`
`[(1 + cos (pi/2-alpha)+ i sin (pi/2-alpha))/(1+ cos (pi/2 - alpha) -i sin(pi/2-alpha))]^(n) = [(2cos^(2)(pi/4 -alpha/2) + i2sin(pi/4 -alpha/2)cos (pi/4-alpha/2))/(2 cos ^(2)(pi/4-alpha/2)-i2sin (pi/4 - alpha/2) cos (pi/4 -alpha/2))]`
`= [(cos (pi/4 -alpha/2)+ i sin (pi/4-alpha/2))/(cos (pi/4-alpha/2)-sin(pi/4-alpha/2))]^(n) `
`[(e^(i(pi/4 -alpha/2)))/(e^(-i(pi/4-alpha/2)))]^(n) = [e^(i(pi/2 -alpha))]^(n)`
`e^(i(n(pi/2-alpha))`
` cos n(pi/2-alpha) + sin n (pi/2 -alpha)`