# Convergence of $1/f(1)+1/f(f(1))+1/f(f(f(1)))+…$

Suppose $f(n+c)>f(n)>1$ for all $c>0$,$n>0$ and that $f(n)\rightarrow\infty$

Must the sum converge?

$$B=\frac1{f(1)}+\frac1{f(f(1))}+\frac1{f(f(f(1)))}+\dots$$

-
did you try anything? any ideas? is this homework? –  yohBS Feb 9 '12 at 7:31
Each of these three can either converge or diverge. Try plugging in some simple functions. Hint: For the harder cases you may need a function that grows particularly slowly. Logarithm is an example of such a function. –  Dejan Govc Feb 9 '12 at 12:30
If $f:\Re^+\to\Re^+$ is subaffine, in the sense that there is a $\beta>0$ so that for all $x>0$, it is $f(x)<x+\beta$ and additionally $f(x)\to\infty$ as $x\to\infty$ (as you have already mentioned), then B converges - Use the D'Alambert convergence criterion. –  Pantelis Sopasakis Feb 9 '12 at 13:36

If $\small f(n)=n+1$ we get the harmonic series which fulfills all your requirements, but diverges...

-
Why did you use \small? –  joriki Feb 9 '12 at 15:23
@joriki : I've got used to it weeks/monthes ago when it fitted my screen-display more smooth than the normal font size. After that I did/do it by routine without explicitely checking/comparing styles again. Did the mathjax-font change a bit in the last time? Maybe my style is no more the best? I'll try out later... –  Gottfried Helms Feb 9 '12 at 15:42
It looks OK, just a bit, well, small :-) –  joriki Feb 9 '12 at 16:26