Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
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Sum of a GP with first term $a$, ratio $r$ ($r \neq 1$): $S_n = a\,\dfrac{r^n - 1}{r-1}$. For large $n$, ratios of GP sums sharing the same $r$ tend to a constant.