@DanielFischer By doing a clever trick :-). Well, we only need to wisely add a parameter and differentiate with respect to it. The rest is an easy job.
@Chris'ssis Oh, I thought you meant a problem. Yes, the solution is very nice, it's yours? Also, how would you go about formally proving that the Riemann sum is equal to the integral (I know it's obvious without proof)?
@Chris'ssis he might mean that these problems you create and solve "are not natural". theory-sticking guys think like that (i fall into that category, sometimes)
@Alyosha The singularity is removable if and only if the principal part of the Laurent series is $0$. It is a pole if and only if the principal part is nonzero and contains only finitely many nonzero terms. It is essential if and only if the principal part contains infinitely many nonzero terms.
If someone uses swear words in chat then they might get flagged by other chat users and subsequently (and automatically) banned.
If someone personally offends a user without using any swear words to the point where the attacked users decides to leave chat for good, nothing happens.
I think th...
@hichris123 Well, some locals (at least four due to starring) that @skullpatrol is well... noise. (S)he usually finds ways to bother people. (S)he has drawn two good users away. Maybe you can talk some sense into him/her.
@skullpatrol Yeah, but it's hard for someone without a math background :-(. Anyway, I'm sure if I went to math uni, I'd be able to finish the first 3 years in one year. (if they let me do that)
@robjohn when you have some time, take a look of this since it's mind-blowing and beyond $$\sum_{n=1}^{\infty} \log\left(1+\frac{1}{n+n^2+n^3}\right)$$
You probably don't know what I'm talking about ... That's the American collegiate sports system ... so the national tennis tournament for college players (many of whom are ex-pats ...)
I am aware. I had thought about it a few hours before I gave in and asked you. :) But I've never before seen a problem on SE generate so many mistaken solutions!!
declining -- you can always cast a reopen vote if the post gets closed.
Also note that all close votes automatically expire after two days.
(and for that matter reopen votes, or any other vote that attempts to reach a threshold -- otherwise, over an absurdly long period of time, say 10 years, e...
@DanielFischer Let $g_n$ be the individual terms in the sum. Maybe one should look at $$\sum_{n\leqslant k\leqslant 2n}g_k(n^{-1/n})$$ to get bad behaviour.
@PedroTamaroff Yes, looking at something like that may be instructive. Can't go thinking about that problem while being online, though, too much distraction.
@Pedro: It is alleged (and I do believe) that the series does converge uniformly.
I confess that early on I cheated and drew some graphs with Mathematica. Usually that's an immediate clue to the answer.
mr eyeglasses: Times have changed. Even the purest of mathematicians now use technology. To discover theorems, verify conjectures that they then try to prove, etc.
If someone uses swear words in chat then they might get flagged by other chat users and subsequently (and automatically) banned.
If someone personally offends a user without using any swear words to the point where the attacked users decides to leave chat for good, nothing happens.
I think th...
