Putnam 2002 A6

Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each $n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of base-$b$ digits of $n$. For which values of $b$ does $$\sum_{n=1}^\infty \frac{1}{f(n)}$$ converge?