Mathematics - Sequence & Series Question with Solution | TestHub

Let $L=\ln a\in (0,1)$. Then $\frac{2L}{1-2L}=\frac{3L}{1-L}$ (valid since $L<\frac{1}{2}$ is needed for convergence). $2(1-L)=3(1-2L)\Rightarrow 2-2L=3-6L\Rightarrow L=\frac{1}{4}$. So $a=e^{1/4}=\sqrt[4]{e}$.