If - is a cube root of unity- then -1-2-3-8-9-32-27-128- - - -

1/2 + 3/8 + 9/32 + ...............= 1/2(1 + 3/4 + 9/16 + ...........∞) = 1/2[1/1 3/4] = 1/2 × 4 = 2. ...

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Cube Root of a Complex Number

If ω is a cub...

Question

If $ω$ is a cube root of unity, then $ω$ + $ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \frac{27}{128} + . . . . . . . . . . . \infty)}$ =

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ω

ω + ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \frac{27}{128} + . . . .)}

^{'} ω^{'}

ω^{(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + . . . \infty)} + ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + . . . \infty)} =

^{'} ω^{'}

ω^{(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \dots \infty)} + ω^{(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \dots \infty)} =

^{'} ω^{'}

ω^{⎛ ⎝ \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + . . . + \infty ⎞ ⎠}

+

ω^{⎛ ⎝ \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + . . . + \infty ⎞ ⎠} =

ω

ω^{⎛ ⎝ \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + . . . \infty ⎞ ⎠} + ω^{⎛ ⎝ \frac{1}{2} + \frac{3}{8} + \frac{9}{32} + . . . \infty ⎞ ⎠} =

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