Evaluate the infinite continued fraction $$\xi = [1,2,3,4,5\cdots] = 1+\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\vdots}}}$$

There is no periodic behavior of this continued fraction, I am not even sure that $\displaystyle\lim_{n \to \infty} [1,2,3,\cdots,n]$ exists or not. Any hints on how to solve this problem?