Consider the fraction $1+\frac{2}{3+\frac{4}{5+\frac{6}{7+\frac{8}{9+\frac{10}{11+\frac{12}{13+...}}}}}}$. This fraction approaches about $1.54$ as you continue it. My question is simple. Does anyone have any idea why this converges to this specific number?
I’ve tried algebraically messing with it, (saying that “$x=1+\frac{2}{3+\frac{4}{5+...}}$” then seeing where it goes.) but to no avail.
I tried to cut off the fraction at different points to try to find patterns in the fractions,( Like cutting off at 7 and solving $1+\frac{2}{3+\frac{4}{5+\frac{6}{7}}}$) but that didn’t work either.
I cannot seem to express it in terms of anything else, either.