Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Must the sum converge?


share|cite|improve this question
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
up vote 3 down vote accepted

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.