I suppose that at least some of these can be considered duplicates. The question is, which of them should be left open and which ones should be closed:
Show that
$$\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i} =0 $$
I've proved that this sequence converges (it is bounded and decreasing). NOW, I need to find a sequence that is bigger than this one and goes to zero. Maybe something using geometric serie of 1/2
Thanks in advance!
I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that
$\lim_{n-> \infty} \frac{H_n}{n} = 0$ where $H_n$ is the nth Harmonic number $= \sum_{i=1}^{n} \frac{1}{...
Find the following limit:
$$\lim_{n\to\infty} \left(1+\frac{1}{2}+...+\frac{1}{n}\right)\frac{1}{n}$$
My intuition says that this goes to zero, because $1/n$ goes much faster to zero than the harmonic series go to infinity, but how can I prove this?
(I have answered one of them before looking for duplicates. Maybe I should have searched first - it is a kind of question which is very likely to have been asked before. On the other hand, the other answers probably did not look for them either.)
Find the following limit:
$$\lim_{n\to\infty} \left(1+\frac{1}{2}+...+\frac{1}{n}\right)\frac{1}{n}$$
My intuition says that this goes to zero, because $1/n$ goes much faster to zero than the harmonic series go to infinity, but how can I prove this?
