@leslietownes Hi! I came across this question: math.stackexchange.com/questions/1220953/a-difficult-integral Seems interesting.

hey. my connection was disrupted for a while. let me look.

oh my.

:P

that's a math version of a rickroll.

The sub u=H-x is obvious

What do you think of the idea to convert the second (extremely) big factor into a generalized binomial theorem?

i'm so tempted to respond and suggest introducing more parameters and then differentiating under the integral with respect to them.

a series expansion might help. i worry that those are real parameters.

Yes but there is a generalized version which is an infinite sum. It is also not that complicated because on summand is only a one.

I found a german wikipedia article but you'll get it. Look at the sum

i see it, but those parameters will appear in the generalized coefficients. and now there will be infinitely many of them.

stuff like that doers sometimes work.

yes this is a disadvantage. an advantage is however, that we can simplify the expression very nicely. but we still have an infinite sum..

@leslietownes Are you still here?

i'm in and out, dealing with a small amount of work stuff.

gone one minute, here the next.

@leslietownes Okay! I wanted to ask you - you are an old timer right? How do you know what a rick roll is ;D?

i'm not that old. the rickroll is old. :)

@leslietownes well that's true, perhaps even older than me haha :P

i thought it was older than it appears to be. i'm surprised we've only had it since 2006-2007? it seems like forever.

@leslietownes To be honest I was suprised to know that it is that "old".My first rickroll was 2, maybe 3 years ago?

it sort of went away for a while. culture moves so fast that other things replaced it.

until that one in the math.se chat a week or to ago, i probably hadn't seen it in years.

@leslietownes Do you mean that peter scholze disproved fermats last theorem?

yes.

yes I had to laugh. I believe he rickrolled everyone who saw his comment

@leslietownes then I rickrolled alexander gruber with the same words and he wrote something like "please be careful with sending links on math.se to other platforms. See [here] for more information." Well guess what this link linked to?

that's clever.

@leslietownes Do you remember that I asked you something about interchanging conditional convergent series? I now found a partial answer I can't understand.

I saw this post. I can't understand reuns comments...

So to be more precise: Why can we only sum n up to log_2(x) and not to infty?

@leslietownes Oh I think I start to understand: We need to find the smallest n for which x^{1/n}<2 holds, right?

@leslietownes So sorry to bother you but I got it! Now I only have the question: Can we always interchange a finite sum (meaning finite terms) and a conditionally convergent sum? (So sorry for spamming :(.)

mm, interchange in what sense?

@leslietownes I think the second answer solves the rest of my question: math.stackexchange.com/questions/1357445/…

yeah.that looks right.

@leslietownes Do you know of other types of convergence? What if a series is neither absolutely nor conditionally convergent? Is it then always divergent? (Consider this a non-rigorous question :)

there are lots of forms of massaging out roughness in divergent things via some averaging process, and saying that the result is 'convergent' in that averaged sense. cesaro summation is the first example that came to mind; there are 'cesaro convergent' series that don't work in the usual sense.

there are a whole lot of these that do not appear to have wikipedia pages. giving them special names seemed to be more popular in the late 19th and early 20th centuy. now people just write down averaging methods and refer to them as whatever they are, and perhaps not as new kinds of 'convergence.'

@leslietownes I wanted to ask if there are like other types of convergence where the series converges but in a special way... But still thank you so much for this list! I know some of these methods but it's very nice to see so many at them at once.

in fourier analysis you sometimes see what they call 'hard approximation' i think, where instead of indexing the terms by c_1, c_2, c_3, c_4, c_5 and doing partial sums in that order, you do sum_{|c_j| > e} c_j and see if the limit exists as you go to zero.

so there you sum the biggest terms first before little ones.

not the easiest formula to 'evaluate' with for specific sequences, but you can do some analysis with it.

as e goes to 0 from the positive side, i should say.

@leslietownes Interesting! So this means there are no "well-known" or "conventional" types of convergence. Also interesting.

well if a space has a natural order on it, people often take limits relating to that. like with partial sums, or improper integrals on R. once you step outside of that it's pretty application specific.

you can invent your own. f(t) converges vitamin-d if lim t->infty f(t) e^(-t)/(cos^(t) + 5) converges in the usual sense.

lots of fun stuff in there.

actually that one's called leslie convergence.

@leslietownes hey I thought it was called vitamin d convergence !!?

you can use e^(-2t) in the numerator.

@leslietownes Nah not good enough. I'll use e^(-t*a), a\in C\setminus{1}.

Uh this gives me some taylor series and mc laurin series vibes

even better.