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I'm a college fresh man on my winter vacation and I'm previewing the part in my next term's Mathematical Analysis that deals with the infinite series. I have therefore learned some tricks for deciding if a series converges or not. Those tricks are quite useful, but only sometimes. When confronted with some very tricky series (perhaps in a rather simple form) they just don't work out very smoothly or even cannot work at all. So my notion is that our tricks for testing convergence are actually quite weak, and that there may well be a good many series whose convergence or divergence hasn't been revealed yet. And thus I'm asking for such an example.

In addition, I want to briefly mention something interesting (my own experience): Before taking an elementary calculus course at the age of 14, I feel very strongly, by my intuition, that $\displaystyle\sum_{i=1}^{\infty}\frac1i$ must be something convergent. I "tested" it on my calculator, which told me that when $i$ turns large (in fact, no more than 50 I believe) the rate at which the sum increases has already been incredibly slow. So even without any proof I thought to myself that it would be a joke if such a thing should diverge. Of course, I was wrong, and I later learned many ways to prove I was wrong. Still, it is no surprise that someone who lacks this part of knowledge and trusts their "mathematical intuition" will, you can almost guarantee, make the same mistake as I did. I simply get this feeling that when we are confronted with a brand new series which none of our tricks can work out (and intuition or technology is of no good help), it is just grossly difficult, or even not quite possible for us to decide its convergence or otherwise. So I firmly believe there are still some tricky ones that still bother us nowadays.


marked as duplicate by MJD, Batman, Dario, Jeremy Rickard, Hans Lundmark Feb 13 '15 at 18:32

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    $\begingroup$ possible duplicate of Are there any series whose convergence is unknown?; also this mathOverflow thread, which I found by doing Google search for "convergence is unknown". $\endgroup$ – MJD Feb 13 '15 at 15:55
  • $\begingroup$ @MJD Oh thank you! that's exactly what I want. well I should have googled it first... $\endgroup$ – Vim Feb 13 '15 at 15:57
  • $\begingroup$ I'm glad I could help. $\endgroup$ – MJD Feb 13 '15 at 15:58

Some series that are converging or diverging depending on the irrationality measure of $\pi$ are: $$ \sum_{n\geq 1}\left|\sin n\right|^{n^3},\qquad \sum_{n\geq 1}\frac{\tan^5(n)}{n}. $$ Many questions on MSE are devoted to similar series.

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    $\begingroup$ Seems to me that as $n$ goes up, $|\sin n|$ will be a kinda random series because of $\pi$'s irrationality, is it true? $\endgroup$ – Vim Feb 14 '15 at 14:15

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