Suppose $f(n+c)>f(n)>1$ for all $c>0$,$n>0$ and that $f(n)\rightarrow\infty$
Must the sum converge?
$$B=\frac1{f(1)}+\frac1{f(f(1))}+\frac1{f(f(f(1)))}+\dots$$
Suppose $f(n+c)>f(n)>1$ for all $c>0$,$n>0$ and that $f(n)\rightarrow\infty$
Must the sum converge?
$$B=\frac1{f(1)}+\frac1{f(f(1))}+\frac1{f(f(f(1)))}+\dots$$
If $\small f(n)=n+1 $ we get the harmonic series which fulfills all your requirements, but diverges...