If ‘ω’ is a complex cube root of unity, thenω13+29+427+……∞ + ω12+38+932+……∞=

13+29+427+....∞=1/31-23=13-2=112+38+932+....∞=1/21-34=4/24-3=2w13+29+932+....∞+w12+38+932+....∞=w+w2=-1

If ‘ω’ is a complex cube root of unity, thenω13+29+427+……∞ + ω12+38+932+……∞=

13+29+427+....∞=1/31-23=13-2=112+38+932+....∞=1/21-34=4/24-3=2w13+29+932+....∞+w12+38+932+....∞=w+w2=-1

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If $‘ ω ’$ is a complex cube root of unity, then
$ω^{(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \dots \dots \infty)} + ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \dots \dots \infty)} =$

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answer is B.

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Detailed Solution

$\begin{array}{rcl} \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + . . . . \infty & = & \frac{1 / 3}{1 - \frac{2}{3}} = \frac{1}{3 - 2} = 1 \end{array}$

$\begin{array}{rcl} \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + . . . . \infty & = & \frac{1 / 2}{1 - \frac{3}{4}} = \frac{4 / 2}{4 - 3} = 2 \end{array}$

$\begin{array}{rcl} w^{(\frac{1}{3} + \frac{2}{9} + \frac{9}{32} + . . . . \infty)} + w^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + . . . . \infty)} & = & w + w^{2} = - 1 \end{array}$