For questions about recurrence relations, convergence tests, and identifying sequences

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Infinite series: existence of power to ensure finiteness

Let $(a_n)_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence converging to $0$. Does there always exist a real number $r\in[0,\infty)$ such that $\sum_{n\in\mathbb{N}}|a_n|^r<\infty$? Does such real ...
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43 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
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109 views

Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
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96 views

How find this sequence such this three condition (AMM Problem 2012)

Prove that for all $n\ge 4$, there exsit integer $x_{1},x_{2},\cdots,x_{n}$ such following conditions, (1):$$x_{1}=1,x_{k-1}<x_{k}<3x_{k-1},2\le k\le n-2$$ (2): ...
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56 views

determining the next random number pseudorandom number generator?

I have given 3 numbers let's say basic example x_0=5, x_1=6 and x_2=2 and modulus p is 7, ...
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25 views

Find $\sum_{n=0}^k \frac{C_n^k A^n B^{k-n} n!}{(\frac{n}{2})!}$.

$$ \mbox{What does}\quad \sum_{n = 0}^{k}{k \choose n}A^{n}B^{k - n}\,{n! \over \left(n/2\right)!}\quad \mbox{converge to ?} $$ I know it converges (I have found an upper bound for it but I can't ...
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64 views

Convergence set of power series

I am trying to find the convergence set of the power series: $\sum_{n=1}^\infty ln\big[1+\big(\dfrac{1}{n}\big)\big](x+2)^n$. So using the ratio test: $\lim_{n\to\infty} \dfrac{|a_{n+1}|}{|a_n|} = ...
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32 views

Consider converge of series[1]

Consider converge of series $$S=\sum_{n=1}^{\infty} \left(\frac{2n-1}{2n+1}\right)^{n\ln n}$$ My tried: Set $u_n= \left(\frac{2n-1}{2n+1}\right)^{n\ln n}$ $\to \sqrt[n]{u_n}= ...
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98 views

All about a failed conjecture.

Some months ago I made the following conjecture - Let $d(n)$ denote the number of divisors of $n $. Then let $N$ be a number such that $d(N)$ divides $N$ . Also let $I= \frac{N}{d(N)}$ which is ...
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98 views

Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
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70 views

Simplify this expression as much as possible?

I am working on a proof and I am stuck on this problem. I have this expression that I want to simplify this as much as possible but I don't get how. I want to try removing the "$...$" using possibly ...
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56 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...
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62 views

Prove $t_{n+1} = \frac{(A-1)t_n^2 + (A-2)\sqrt{A}t_n}{A(t_n + \sqrt{A})}$ converges for $A > 1$

I want to show that the following sequence converges, given that $A$ is any real number greater than $1$: $$t_1 = A-\sqrt{A}$$ $$t_{n+1} = \frac{(A-1)t_n^2 + (A-2)\sqrt{A}t_n}{A(t_n + \sqrt{A})}$$ ...
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35 views

Prove that when converge, the following expansions are equal

Prove $f_1(x)=f_2(x)=f_3(x)$ when converge. $$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$ $$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom ...
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32 views

“In between” for Lagrange's theorem for series

Everywhere I look, Lagrange Remainder Theorem is stated as Let $f$ be $k +1$ continuously differentiable, …, then there is a number $c$ in between $x$ and $a$ such that $$R_k(x) = ...
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77 views

The sum of an infinite series

I'm working on a probability question and I want to find the precise summation of: $$\sum_{p=1}^\infty \frac{n^{p-1} - \sum_{x=0}^{p-2} n^x}{n^p}$$ Where $n$ is an integer greater than $1$. More ...
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38 views

Proving that a convergent infinite series implies convergent part of that series and vice versa

I am asked to prove the following theorem: $\sum_{n = 1}^\infty a_n$ converges if and only if $\sum_{n=N}^\infty$ converges. I am not sure if I have done this correctly: Suppose that ...
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49 views

Sum of the series problem

I was doing a programming problem.There i faced the difficulty of solving two series. The equation which i was asked to solve was $Z_n+Z_{n-1}+2Z_{n-2}$. Where $Z_{n}$ is given as $P_{n}+S_{n}$. ...
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Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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65 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
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Limit of a quotient involving the gamma function

This is a continuation of another question I asked. It seems only incidentally related to statistics, so I figured it would be better separate. (Also, I'm not able to make nearly as much progress on ...
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100 views

Summary of divergent series summation methods and relations between them?

There are a number of methods of assigning sums to series that do not necessarily converge, e.g. Cesàro summation, Abel summation, Ramanujan summation, etc. (There is also the trivial method of only ...
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25 views

Renormalization operator R

In a Robert Devaney's book ("an introduction to chaotic dynamical systems") is approached the quadratic map $F_\mu=\mu x (1-x)$. He introduce the renormalization operator R. R is a function of ...
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85 views

Analysis Question based on Series

I've been having some difficulty solving this question especially with understanding how to comput the specific subesequence specified and why they relate to solving this question. The question is: ...
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62 views

Constant sequence limit at infinity

Obviously the sequence $\left(\log (n)\sin(n\, \pi)\right)^{\infty}_{n\,\in\,\mathbb N}$ is $0$ for every $n$ but when I would like to take a limit of this sequence as $n$ approaches infinity I get ...
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Conditional convergency of complex series

The Riemann theorem states that rearrangements of conditionally convergent series of real numbers may have each sum from $[-\infty,+\infty]$. Are there any similar theorems about series of complex ...
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58 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
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Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
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Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
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35 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
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70 views

What to compare this integral with to prove its convergence?

I have the following integral: $$ \int_e^\infty \frac{1}{x \ln^2 x} \mathrm{d}x $$ What should I compare it with in order to prove its convergence? I know how to calculate the integral and find a ...
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66 views

A property of a sequence of summable sequences

Let $(a_n)$ be a summable sequence of positive real numbers then we can find a sequence $w_n\to \infty$ such that the sequence $(a_nw_n)$ is still summable. This property has been asked and answered ...
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72 views

Help with Identifying Cipher

Anyone know what type of cipher this might be? 222132143135533 3335521 2214124313 135 35135 353314142412 31253435 313135 1434 2225313554 135 2425333513 351314333545341444 351314333545341444 ...
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nonlinear sequence to sequence transformations

i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation?
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Divergence of a double sum

I wanted to prove the divergence of the following sum $ \sum\limits _{n=2}^\infty((-1)^n\sin(b\log(n))g(n)$. I expanded the sine function and rearranged the terms to get $\sum\limits_{i=1}^\infty ...
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62 views

Conditions for Martingale

Let $X{}_{1},X_{2},\ldots$be a sequence of independent RV and let $f{}_{j}$be continuous functions. Let $S{}_{0}=1$ and $S{}_{n}=\sum_{j=1}^{n}f_{j}\left(X_{j}\right)$. Find a necessary and sufficient ...
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DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
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Remainders in Alternating Series / Remainders in general

So there's a little part I get stuck on when I'm trying to find the remainders, i'll post two simple problems with my work and my answer compared to the book answer. Please help me out and tell me ...
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Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
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89 views

Uniform convergence of function series using Dirichlet criterion?

In some exercises for an analysis course, we are asked to prove that the function series $\sum f_n$ converges uniformly on $[-1,1]$, where $$ f_n:[-1,1]\to \mathbb{R}:x \mapsto \frac{x^n sin(nx)}{n} ...
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686 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
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Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
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Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
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Sequence with a contraction mapping among others

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
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Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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96 views

Question about the double limit of a sequence

Let $a_{mn}$ be a double sequence. Then under what conditions do we have $$\lim_m\lim_na_{mn}=\lim_n\lim_ma_{mn}$$ In particular, if the sequence is positive, and for each fixed $n$, the limit exists ...
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49 views

Finite Sum of Zeta Functions

Let $T(n) =$ the zeta function evaluated at $x = n$. Is it possible to find the sum Sigma ($k =1$ to $n$) of $T(k)$ ? via impropal integrals? closed form? I need this to evaluate a sum involving zeta ...
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144 views

Maclaurin series of $\sin(2\pi x)$

Find the Maclaurin Series for $$f(x) = \sin ( 2 \pi x )$$ using the definition of a Maclaurin series. If $f(x) = \sum_{n=1}^\infty c_{2n+1}x^{2n+1}$, give $c_{2n+1}$: ...
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48 views

Partial limits of recursive sequence

Let $\alpha \in \Re$. Find all the parial limits of the sequence $a_n$ given: $a_1=\alpha$ $a_{2n}= \frac{a_{2n-1}}{2}$ $a_{2n+1}= \frac12 +a_{2n}$ I've tried to show that $a_n$ has two ...
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Finding every set of the five consective terms of an arithmetic sequence such that the four terms of them are squares

Question : Can we find every set of the five consective terms of an arithmetic sequence such that the four terms of them are squares? Motivation : It has been known that four consecutive terms in ...