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Consider the following series:

$$\sum_{n=1}^{\infty} \frac{3^{2^n}}{2^{3^n}}$$

I believe that this series converges, but I don't know how to prove this.

Thanks in advance!

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Try the ratio test... – Sanath K. Devalapurkar Feb 14 '14 at 2:08

Let $a_n=3^{2^n}/2^{3^n}$. Since $\lim_{n\rightarrow\infty}a_{n+1}/a_n=\lim_{n\rightarrow\infty}3^{2^n}4^{-3^n}=0$, from D' Alembert's theorem we know the sum of $a_n$ is convergence.

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