Consider the following recurrence relation in two variables: $$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$ for positive integers $a$ and $b$, with the boundary conditions $f(0,b)=0$ for integers $b>0$ and $f(a,0)=1$ for integers $a\ge0$.

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

Numerically the convergence is slow.