Self-contained infinite series – Mathematics Stack Exchange

What do you call such an infinite sequence? $ (x_n) _ {n in Bbb N}: forall a = (x_i, x_ {i + 1}, x_ {i + 2}, …, x_ {i + j}) exists b = (x_k, x_ {k + 1}, x_ {k + 2}, …, x_ {k + j}); i not = k: a = b $

For example, the 2-adic rating of 2n, https://oeis.org/A001511

In other words, every part of the sequence has a copy of itself in the sequence. In fact, each part would have an infinite number of copies of itself. I thought that was a fractal sequence, but that's not the case.

Generalization of Menger's theorem to infinite graphs

Aharoni and Berger generalized Menger's theorem to infinite graphs: for every digraph and subset A and B, there is a family F of disjoint paths from A to B and a set separating B from A and from a vertex of each path in F consists.

But what I need is a criterion (at least one useful enough condition) that tells me when there is a family of non-contiguous paths from A to B covering all A's 3 vertices.) What's Known?

Prove: If finite automata M with k states accept a string with at least k signs, then the language L (M) is infinite

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Recursion – Generate a large (possibly infinite) list of formulas from a group

Here's a little background, I have a finite list of linear formulas in 6 variables and 6 linear transformations in those variables. Of course I can apply these transformations with /. to my current formulas to get new formulas. I want to automate the following process in Mathematica:

  1. apply all transformations to my current formula list
  2. Take the union of my new and old formulas to get a new list
  3. Apply step 1 to this new list
  4. After n iterations, stop steps 1-3 for an integer n
  5. Sort my final list canonically, taking all formulas naturally into account

Is there an easy way to do this in Mathematica? I'm not familiar with the program at all. Incidentally, this is not a homework assignment, I have only one guess in mind and would like to skip pages and pages of manual work on which I may make mistakes. If it helps at all, these formulas form a group.

pi – Existence of infinite series for $ ln (π) $

We all know that there are representations for many series $ π $,

But here I want to know if there is an infinite series for $ lnπ $,

From the Valais formula (in the form of logs) we can see an apparent.

I also know the following series

begin {align}
ln (1+ (7 pi / 22-1)) & = – sum_ {n = 1} ^ infty frac {(- 1) ^ n} {n} (7 pi / 22-1) ^ n
end

But I want to know if there is a "simpler" with rational terms, with possibly few tradable irrational terms?

Infinite divisibility of log normals – MathOverflow

TL; DR: What is the low point of a piece of logarithmic normal distribution?

We know that logarithmic normals are infinitely divisible. What would be the low point of a root of lognormal?

More precisely, let us assume that $ X $ is a log normal. Given an integer $ k> = 2 $, we now, since there is $ X_1, …, X_k $ so that:

$$ mathcal {L} (X) = mathcal {L} ( sum limits_ {i = 1} ^ {k} X_i) $$

From where $ mathcal {L} $ denotes the distribution of a random variable.

$$ Question: $$ Is there a way to simulate directly from $ X_i $Law?

Computer Architecture – Can we also represent $ sqrt {2} $ with infinite bits in the mantissa?

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Abstract Algebra – Show that the set v of all vectors (defined as directed line segments) forms an infinite Abelian group with vector addition as a composition

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Limit of an infinite sum

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The above is from Maple 2019.1. Is there a way to achieve the same result with MMA12?

Tries

Limit[Sum[Sqrt[1 + k^2/n^3] - 1, {k, 1, n}], {n -> Infinity}]

which did not work. The best I can get is a numerical approximation.

With[{n = 10^200}, NSum[Sqrt[1 + k^2/n^3] - 1, {k, 1, n}]]

0.166667

which is informative,