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Jun
29
comment Is there something between summation and integration?
this is answering a very different question actually, but it might be of interest
Jun
28
comment How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization
@AsafKaragila I specify this was done with zeta function regularization. Would it be more precise to ask for the analytic continuation $\zeta(0)$ ?
Jun
28
comment How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization
Could you prove Ramanujan summation as Euler-Maclaurin to 1st order?
Jun
28
comment How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization
hmm? that seems plausible that $\sum (0+2) = \sum (1+1)$ and yet $\sum (0+2) = 2 \sum 1$.
Jun
28
comment How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization
@AsafKaragila this false statement has its own wikipedia article
Jun
28
comment What we get if we add 1/2 infinite times
@Ant There is nothing wrong with this kind of experimentation. Euler did it, Physicists do it routinely (via their lack of rigour). He just needs to understand these series come with at the expense of associativity. We can't invoke $(a+b)+c = a + (b+c)$ infinitely many times and always get away with it. Watch his parentheses carefully.
Jun
28
comment What we get if we add 1/2 infinite times
@Ant his question seems to be whether these divergent series identities can be found via elementary rearrangements.
Jun
25
comment Integral relations in Fricke and Klein
@glebovg I think they want to give you the option of not awarding any bounty at all. In the case nobody answers the way you wanted.
Jun
24
comment Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?
I really like your numerical approach to this problem.
Jun
24
comment Arnold Trivium Problem 39
@Potato since the Gauss integral does not change under deformations, we can deform $A$ into a straight line or (even easier) flatten $B$ into a circle, where we going around twice. Then the integral might be doable by hand. This is line when we integrate complex functions using Cauchy Residue Formula.
Jun
24
comment Arnold Trivium Problem 39
@Potato I wouldn't dream of doing the integrals by hand... unless you truly enjoy long and tedious integrals :-)
Jun
24
comment Arnold Trivium Problem 39
I have to step away, but I can do a numerical calculation... just to see!
Jun
24
comment Arnold Trivium Problem 39
e.g. Gauss Linking Number Revisited (Ricca + Nipoti) Journal of Knot Theory and Its Ramifications
Jun
23
comment Integral relations in Fricke and Klein
@glebovg i added more details... why didn't you give me the 100?
Jun
19
comment Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$
@jack any reason in particular $4$?
Jun
15
comment Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.
@1233dfv I was stuck there too. Either you add all the $n$ which are perfect squares or finally I decided $n = k^2$.
Jun
15
comment Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.
is this the Cesaro sum or something? I don't think spacing the non-zero values changes the values of $c_n$.
Jun
15
comment Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.
no doesn't $\sum |c_n|^2 = \sum \frac{1}{n} = \infty$ ?
Jun
14
comment Two sets $X,Y \subset [0,1]$ such that $X+Y=[0,2]$
It was Problem 3, Part 1 at CIIM 2010 Rio de Janeiro
Jun
14
comment Limits and Series in Smooth Infinitesimal Analysis
@StefanPerko I looked into it further. With these type of "nilpotent" infinitesimals, $\epsilon$ is not really "small" it just vanishes to a certain order. More specifically, the Intermediate Value Theorem is false. Likewise, many of the series and limit definitions you have defined may not work in SIA.