If '$\omega$' is a complex cube root of unity,then $\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 }+\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 }  +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty  \end{pmatrix} }=$

$\dfrac{1}{3}+\dfrac{2}{9}+\dfrac{4}{27}+......+\infty$ (infinte G.P.)

$S_\omega = \dfrac{a}{1-r} =1$

$\dfrac{1}{2}+\dfrac{3}{8}+\dfrac{9}{32}+......+\infty$ (infinte G.P.)

$S_\omega = \dfrac{a}{1-r} =2$

We know,

$1+\omega+\omega^2=0\\ \therefore\, \omega^1+\omega^2=-1$