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      Questions tagged [sequences-and-series]

      The tag has no usage guidance.

      2
      votes
      1answer
      85 views

      Where to find the proof of a property?

      I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
      0
      votes
      0answers
      38 views

      Infinite products from the fake Laver tables-Now with no set theory

      We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
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      votes
      0answers
      44 views

      What is the Laurent series of z+(1/z)? [closed]

      What is the Laurent series of z+(1/z)? Is it just the series itself?
      6
      votes
      1answer
      186 views

      Asympotic density of a very simple sequence

      Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite. I'm actually even more ...
      1
      vote
      0answers
      63 views

      Existence of limit in a recurrence equation: $\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}$ [migrated]

      Let be $\boldsymbol{\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}}$ a recurrence equation with known $\alpha_0$ and $\alpha_1$. How do you prove that $\lim_{n\to\infty}\alpha_n$ exists? Note that ...
      0
      votes
      1answer
      184 views

      Estimate $\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$ [closed]

      Suppose $a>1,b>0$ are real numbers. Consider the summation of the infinite series: $$S=\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$$ How can I give a ...
      0
      votes
      0answers
      103 views

      Infinite products of complex numbers or matrices arising from rank-into-rank embeddings

      I wonder what kinds of closed form infinite products of matrices, elements of Banach algebras, and complex numbers arise from the rank-into-rank embeddings. Suppose that $\lambda$ is a cardinal and $...
      1
      vote
      1answer
      69 views

      How to obtain a product-to-sum identity for the sinc function?

      We know that $$\text{sinc}(x)=\prod_{n=1}^\infty\cos\left(\frac{x}{2^n}\right)$$ and for some truncated $k$ we can write the following product-to-sum identity: $$\prod _{n=1}^k \cos \left(\frac{x}{2^n}...
      0
      votes
      0answers
      84 views

      Sum of products of binomials satisfies recurrence relation?

      i need to know if the sequence $(a_n)_{n \geq 0}$ defined by $$a_n =\sum_{s=3n}^{16n} C_{15n}^{s-3n} \; C_{14n}^{s-2n} \; C_{12n+s}^{12n} $$ satisfies a recurrence relation ( type sequence Apery) or ...
      2
      votes
      0answers
      248 views

      How to show that the sinc function series $\sum_{n=-\infty}^\infty\text{sinc}(x+n)$ is equal to pi for all $x$? [closed]

      We know the very well-known identity: $$\sum_{n=-\infty}^\infty\text{sinc}(n)=\pi.$$ But how to show that $$\sum_{n=-\infty}^\infty\text{sinc}(x+n)=\pi, \qquad \forall x?$$
      -2
      votes
      0answers
      23 views

      How to prove sinc function series? [migrated]

      I am not sure how to prove this series for the sinc function: $$\text{sinc}(x)=2\cos\left(\frac{x}{2}\right)\sum_{n = 0}^\infty\frac{(-1)^n x^{2 n}}{2^{2 n + 1}(2 n + 1)!}.$$ Is there an elementary ...
      2
      votes
      1answer
      56 views

      Decaying of a certain ratio of binomial sums

      Consider the two sequences $$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$ and $$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$ QUESTION. Is this true? $$\frac{a(n)}{b(n)}\...
      0
      votes
      1answer
      106 views

      Simplify a double summation involving binomial coeficient

      $$T(N,K)=\sum_{i=2}^{K}\sum_{j=2}^{i}(-1)^{i-j}\binom{i}{j}\frac{j^{N+1}-1}{j-1}$$ Is it possible to evaluate the sum for $K=10^7$ efficiently. If we manage to remove one of the sums, it will be ...
      -1
      votes
      0answers
      42 views

      Convergence of a recursion in $L^1(0,1)$

      Let $\mu_0>0$, $a>0$, $b>0$, and $f(t)$, $g(t)>0$, $p(t)$ be some continuously differentiable functions over $\mathbb{R}$. I am looking for various tools to study the stability of the ...
      12
      votes
      2answers
      465 views

      Integer but not Laurent sequences

      Are there any sequence given by a recurrence relation: $x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a positive Laurent Polynomial, satisfy: if $x_0=\cdots=x_{n-1}=1$, then the sequence is only ...

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