Consider the following recurrence relation in two variables.

$$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$

with the boundary conditions $f(0,b)=0$ for $b>0$ and $f(a, 0)=1$.

I believe that $f(1,n)$ converges 1 as n tends to infinity. However I can't see how to prove it.

Numerically the convergence is slow.