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

Must the sum converge?

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

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    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) and additionally $f(x)\to\infty$ as $x\to\infty$ (as you have already mentioned), then B converges - Use the D'Alambert convergence criterion.2012-02-09

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

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    It looks OK, just a bit, well, small :-)2012-02-09