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Jul
19
comment How to replace addition with multiplication to find the next integer value?
@siméon: hadn't just this been the point/the nerve of the couriosity of the OP?
Jul
18
comment List of divergent series and their summations
what about wikipedia? There is already a certain list. Why not extend that list?
Jul
15
comment Looking for a verification or refutation my attempted proof of why the Collatz conjecture is probably false.
Have you ever tried to verify the conjecture that $g(x)=2^x$? I've just rund some tests and tried for instance $g(20)$ so 20 steps to reach 1 (or : how much numbers can be generated if you start by n=1 and do the inverse collatz-transformation for 20 steps?). For that example I got $g(20)=321$ or $g(20)=322$ (depending on whether we count the cycle at 1 as well) This comprises from $71$ unique odd numbers plus $250$ additional even numbers as even multiples of the odd unique numbers (only cofactored by perfect powers of 2). See my answer-box for a listing of the odd numbers
Jul
12
comment What is the direction along the edge of a circle called (in English and by chance German)?
@Grantwalzer: Well, now I hope it comes out well for your problem and the ideas were helpful in this or that way.
Jul
11
comment What is the direction along the edge of a circle called (in English and by chance German)?
@Grantwalzer: Also "versetzt" is a better term than "verschieben" in this case - because "verschieben" is somehow felt to be a continuous motion while "versetzt" emphasizes the discreteness of the motion. However - why the limitation on "wachsend" (growing). For a general sentence is "verändert" better, because it doesn't imply only one direction of change of radius. Why should the problem for the reader be specialized to that case of "growing"/"expanding"?
Jul
11
comment What is the direction along the edge of a circle called (in English and by chance German)?
Perhaps "Veränderungen des Radius führen zu Verschiebungen der Punkte auf dem Rand (die Verschiebungen sind nicht-stetig)" . This has also a causal notion. To remove this causal notion I'd say "Veränderungen des Radius gehen einher mit Verschiebungen der Punkte auf dem Rand (die Verschiebungen sind nicht-stetig)". The term "Rand" is here a short form for "auf dem Umfang" ("on the circumference"), and possibly is a bit outdated/today too little technical.
Jul
8
comment A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
I have a Pari/GP-routine which makes it really simple. It uses sumalt , a procedure for divergent summation (based on Zagier's ideas) on formal powerseries. Possibly in Math'ica this is possible as well. I can post it here if you're interested. (Or search for "stieltjes" and my user name, possible I've already posted that routine here or in MO)
Jul
8
comment A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
I assume, the $\gamma$ are the Stieltjes-constants? (I just tried to confirm it numerically using $n=100$,$n=1000$, $n=10000$ and $n=200000$ and seems to approximate Stieltjes $\gamma_1$ and $\gamma_2$
Jul
4
comment Algorithm for tetration to work with floating point numbers
Perhaps this (go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf) is an interesting essay, although it is extremely amateurish (one of my earlier tries to put numbertheoretic puzzles into shape - I had even difficulties to refer to functions having powerseries in the usual way) - I should rewrite this. But it might give a good impression for a first read.
Jul
4
comment Algorithm for tetration to work with floating point numbers
Hmm, I'm not much active on this this days. Look at my oldest postings, I've called that "matrixmethod" (this is perhaps the most fruitful searchable term). And in the beginning I did not know that this all was already known with the term "Carlemanmatrix". There is also some literature on this, look for Aldrovando (the link to the ArXiv by D. Geisler), Eri Jabotinsky, S.C.Woon (perhaps I can come back to this later with more information - we have it extremely hot today which makes it uncomfortable to sort out links...)
Jul
1
comment Super root function
If something useful should exist, it's rather the (analoguous) pentation-"squareroot". Unfortunately the pentation is not remotely so far developed as the tetration. Maybe you find something in the "tetration-forum" math.eretrandre.org/tetrationforum
Jun
23
comment The limit of the alternating series $x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb$ as $x \to 1$
Didn't you see the formula 3.18 at page 100 of your link? They say the following . Amplitude is about (pg 100 formula 3.18): $ 2/\log(2) \cdot | (\Gamma(\pi \cdot I/ \log(2)))| \approx 0.00274922168400 $ and this is alternating around $0.5$. So there is nothing with $0.333...$ and $0.6666...$ in that article...
Jun
23
comment The limit of the alternating series $x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb$ as $x \to 1$
I gave it a +1 although it seems that the result contradicts other (and my own) results because of the lot of serious work in this answer...
Jun
23
comment The limit of the alternating series $x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb$ as $x \to 1$
Dear Abdou, please see also my graphics in the (same )question at MO from february: mathoverflow.net/questions/198665/…
Jun
23
comment The limit of the alternating series $x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb$ as $x \to 1$
You may also look in this identical Mathoverflow question mathoverflow.net/questions/198665 ; some -I think interesting- additional info is in my answer. See mathoverflow.net/questions/198665/…
Jun
17
comment How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
Will, I've just seen, that your $\lambda()$-function down to the Abel-function $\alpha()$ can directly be determined by the use of the log of the Carlemanmatrix F for the function $f(x)$ . I did never know that this is such an easy relation between that two computation-concepts. See the end of my long answer.
Jun
16
comment More numbers between $[0,1]$ or $[1,\infty)$?
As @columbus8myhw mentioned in his comment: which number in the second set is mapped from the 1 in the first set?
Jun
10
comment Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
I've got an answer at MO jun 2015: mathoverflow.net/a/208900/7710 which resolves the problem.
Jun
7
comment Cesàro means of divergent series
Hmm, also I suspect that at the date of my answer I might not have thought your question correctly to the end because I end with consideration of the sums, not means. I'll look again at this when I've more time
Jun
7
comment Cesàro means of divergent series
Hmm, I'm not sure at the moment, but isn't the mean not just the $n$'th partial (Cesaro) sum divided by $n$? Then a finite Cesaro-sum $s_n$ for $n \to \infty $ should go to $\lim_{n \to \infty} s_n / n = 0$