<s>[INST] convergence series geometric test Prove if this converges:
$$\sum_{n=1}^\infty \frac{2^n+3}{3^n-1}$$
pf: using geometric
$$0 < \frac{2^n+3}{3^n-1} \leq \frac{2^n + 2 \times 2^n}{3^n-\frac{3^n}{2}} = \cdots $$ and so on
I know how to do the rest but my question is that where in the world did my teacher get 
$$\frac{2^n + 2 \times 2^n}{3^n-\frac{3^n}{2}}$$
 [/INST] Ratio test:
$$
\frac{\left(\dfrac{2^{n+1}+3}{3^{n+1}-1}\right)}{\left(\dfrac{2^n+3}{3^n-1}\right)} \to \frac 2 3 \text{ as }n\to\infty,
$$
so the series converges.
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