1.

\lim _{n \rightarrow \infty} \frac{1+\frac{1}{2}+\frac{1}{2^{2}}+\ldots+\frac{1}{2^{n}}}{1+\frac{1}{3}+\frac{1}{3^{2}}+\ldots \ldots+\frac{1}{3^{n}}}

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