CONVERGENCE in the PRIME RATIOS

The (n + 1)th Fibonacci number in a ratio with the nth Fibonacci number has a limiting value equal to the golden ratio. Does the (n + 1)th Prime in a ratio with the nth Prime converge to a limiting value equal to some significant number in mathematics? It looks like 2 over 1 equal to 2 is the highest value of such a ratio....where the rest of these ratios fall between 1 and 2 and they jump around and do not converge on any significant number in math. I did a search for the convergence of such ratios on Bing and I did not see anything there on any kind of mathematical law of convergence of this sequence of ratios.....but someone has figured it out....but I have never heard of it.