How to evaluate this infinitely nested fraction? $$ 1 + \frac{1}{1 + \frac2{1 + \frac3{1 +\cdots}}}$$
I tried to define some $g(x)$ such that $$g(x) = 1 + \frac{x}{1+\frac{x+1}{1+\frac{x+2}{1+\cdots}}}$$ and solve the equation $$g(x) = 1 + \frac1{g(x+1)}$$
I was unsuccessful in solving it, so is there some other, better way at approaching this problem without bringing in functions? Or perhaps is there an easier way to solving the equation I was stuck on?
