Let $\displaystyle p_n$ represent the n-th prime number. Determine what $\displaystyle \lim_{n\rightarrow \infty}\frac{p_{n+1}}{p_n}=?$. Does it even have a convergence?

Notice the following

$\displaystyle p_n<p_{n+1}<2p_n$ (Betrand's Conjecure)

thus,

$\displaystyle 1<\frac{p_{n+1}}{p_n}<2$

Thus, by the limit limitation theorem,

$\displaystyle 1\leq \lim_{n\rightarrow \infty}\frac{p_{n+1}}{p_n} \leq 2$ (I, of course, assumed the limit exists).

My guess is that the limit is one.