If $‘\mathrm{\omega }’$ is a complex cube root of unity, then

${\mathrm{\omega }}^{\left(\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\dots \dots \infty \right)} + {\mathrm{\omega }}^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\dots \dots \infty \right)}=$

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

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

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