# Tagged Questions

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### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
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### An increasing sequence of real numbers [on hold]

This was first posted to SE, but now I think its better to be posted here. For what positive real numbers $\alpha$, the sequence $a_n = \frac{\lfloor n\alpha\rfloor}n$ is (not necessary strictly) ...
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### A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
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### Is $\sum\limits_{n=0}^\infty x^n / \sqrt{n!}$ positive?

Is $$\sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0$$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
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### Asymptotic upper bound for recursive function $f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$

I'm looking for an asymptotic upper bound for the function f(x), which is recursively defined as follows: $$f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$$ with $f(1)=1$. I am pretty ...
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### A term for sequences whose mean is defined?

This may be an extremely stupid and elementary question, but is there a name for sequences $\{a_i\}$ such that $\lim_{n\to\infty} \left( \frac{1}{n}\sum_{i=1}^{n} a_i\right)$ exists? This seems to be ...
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### The maximum lengthed sequence of prime numbers with certain conditions (denizens)

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P}$; and it satisfies the following condition; ...
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### Number of binary sequences in which the number of $(1, 1)$ and $(0, 0)$ is prespecified

Consider a binary sequence $\mathbf{a}_n$ consisting of 1s and 0s. Let us denote by $f(\mathbf{a}_n)$ the number of $(1, 1)$ and $(0, 0)$ in $\mathbf{a}_n$; I am not sure whether there is a formal ...
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### Generating a series representation for the inverse of the operator $f(f)$

I was considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
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### How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question. For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...
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### Asymptotic behaviour of a sequence

Hello, I am interested in some kind of sequence that are "not finitely recurrent". Let $a_i$ be a sequence taking values in $\{0,1\}$. Consider the sequence $(u_i)$ such that $u_0=1$, and for any ...
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### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
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### Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...
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### Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
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### Elementary treatment of elementary functions in constructive math

I would appreciate a reference to constructive math literature with elementary proofs that elementary functions are locally non-constant (i. e. densely apart from any real in any interval with ...
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### Any formula for the partial sum of a remainder series?

Let $N \ge 1$ be an integer, and there is a series $\{ N \mod 1, N \mod 2, ... , N \mod i, ... \}$. Obviously when $i \gt N+1$, the series will become $\{N, N, N, ..., \}$. So only take $i \le N$ ...
How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?