I need help in solving the limit below:

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> $$\lim\_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$
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What I've done is to simplify the upper part to: $$\frac{2^{n+1}-1}{2^n}$$

Any hints or solutions will be greatly appreciated.