How to define a nested function like this?

For example, I want to calculate the limit of $\frac{1}{1+\frac{2}{1+\frac{3}{1+\ldots}}}$. So I want to define:

$f(1)=1$

$f(2)=\frac{1}{1+2}$

$f(3)=\frac{1}{1+\frac{2}{1+3}}$

Then I calculate the $n\to\infty$ limit of $f(n)$.

However, I do not know how to define such a funcion in Mathematica. It seems we cannot change the parameter in Nest.

Answer

You can use Fold instead:

f[n_Integer] := Fold[#2/(1 + #) &, n, Reverse@Range[n - 1]]
f[3]

$\frac{1}{1+\frac{2}{1+3}}$

It not very useful analytically, but it allows you to invoke the CPU gods:

f /@ Range[50] // ListLinePlot[#, PlotRange -> All] &