I was messing around on desmos and accidentally mistyped and wrote "$1+1/(2-1/3+)$" instead of "$1+(1/2)-(1/3)+$". Anyways, I thought I might continue that trend and try to see what that infinite series $$1+\frac{1}{2-\dfrac{1}{3+\dfrac{1}{4-\dfrac{1}{5+\dfrac{1}{6-\dfrac{1}{7+\cdots}}}}}}$$ would look like. I found that this series of fractions converged rapidly on a value that is something like $1.59049113525$ after 10 iterations.
Could someone tell me what this fraction is and what it eventually approaches?