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?
@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 :(.)
@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.
In mathematical analysis, a summability method is an alternative formulation of convergence of a series which is divergent in the conventional 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.
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.