Questions tagged [convergence-divergence]
Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.
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Cannot prove these two statements about Bernoulli sequence
In the 8th chapter of An Introduction to Probability Theory and its Applications. Vol 1 (by William Feller) there are two problems that I'm stuck with, and since the two are interconnected, I decided ...
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How do we know this series diverges?
I’m trying to solve some problems relating to convergence, and I came across this problem:
$$ \sum_{i=2}^{\infty} \frac{1}{(\log (\log i))^{1+\varepsilon}} = +\infty , \,\,\, \varepsilon > 0$$
My ...
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Proof a sequence is convergent
Given 2 sequences $\{x_n\}$, $\{y_n\}$ such that:
$${y_n^{2} \leq \frac{1}{n} + {x_n}{y_n} \sqrt[3]{x_n}}$$ with ${\forall} n \in$ $ \mathbb{N}$.
Suppose that ${x_n} \to 0$. Prove that $\{y_n\}$ ...
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Does the given sequence converges or diverges? [closed]
If b>1 and is a real number then does the sequence converge or diverge. I intuit that it diverges because b^n grows faster than (n!)^1/2?
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Show that $L^p$ convergence is unique almost surely
It'd be of great help if someone could double-check if my proof is correct/rigorous.
Problem: If $X_n\to X$ and $X_n\to Y$ in $L^p$, then $X=Y$ almost surely
My proof: We have that $\lVert X_n -X \...
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How can I show that this limit exists and equals 1? [closed]
$$
\lim_{n \to \infty} \frac{(n+1)^2}{2^{2n+1}}
$$
I am stuck on this part after I applied ratio test to the series.
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Proving that convergence in RKHS implies pointwise convergence without using reproducing property
Let $(\mathcal{H}, \mathcal{K})$ be a reproducing kernel Hilbert space and denote $\mathcal{K}_x := \mathcal{K}(x, \cdot)$.
Is there a simple way to prove $f_n \to_\mathcal{H} f$ (shorthand for $\|f_n ...
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Example of a sequence $x_n$ which converges to $0$, $n(x_n-x_{n+m})=O(1)$ but $(x_n-x_{n+k}) \notin \ell^1(\mathbb{N})$
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $x_n \to 0$ as $n \to \infty$. I want to know if the following conditions are equivalent or if there is one weaker.
$n(x_n-x_{n+...
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Show the convergence of a sequence of random variables
Consider some discrete random variables $X_1,X_2,...$ each with support $A$. Take another sequence of random variables $Y_1,Y_2,\dots$
Given $x\in A$, assume:
A1: $\Pr(X_t=x| X_{1},\dots, X_{t-1}, Y_1,...
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Just as Fibonacci addition approaches phi with a(n-1)*phi = a(n), Fibonacci multiplication approaches phi with a(n-1)^phi = a(n). Can we extend?
Fibonacci addition converges on the golden ratio between consecutive values, i.e. a(n)/a(n-1) = phi. Example: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... 55/34 rounds to 1.618.
I realized today that Fibonacci ...
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Find convergence of $\sum_{n=1}^{\infty}\frac{1}{n^2-ln(n)}$
$$\sum_{n=1}^{\infty}\frac{1}{n^2-ln(n)}$$
I was compering it to $$\sum_{n=1}^{\infty}\frac{1}{n^2}$$
and I had limit $$\lim_{n\to \infty}\frac{a_n}{b_n}$$
where $${a_n}=\frac{1}{n^2-ln(n)}$$ and $${...
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Limit of expectation of reciprocal mean of uniforms
Suppose we have a sequence of iid unif$(0,1)$ random variables. I want to know whether or not the sequence of $\mathbb E[\frac1{\bar X_n}]$ converge to 2 (since $\bar X_n$ strongly converges to $\...
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How to prove that an infinite series does not converge to $0$?
I am now considering the following infinite series.
$$ \lim_{k \to \infty} \sum_{j=1}^{k} \frac{1}{j(j+1)} \left\{\exp\left(\frac{2 \pi i}{n} \right)\right\}^{j} $$
Here, $n \geq 2$ is a natural ...
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Find Convergence [duplicate]
$$\sum_{n=1}^{\infty}(n(\sqrt[n]{e}-1))$$
I get rid of sqrt and made e^1/n and for me it looks like Dahlembert principle,but I might be wrong and so confusing by this...I'll appreciate someone's help
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necessary and sufficient condition about weak convergence on N (the set of natural numbers)
I would like to show that if all natural number $k$ satisfy that
$\displaystyle\lim_{n\to\infty}\mu_{n}(\{k\})=\mu(\{k\})$, then $\{\mu_n\}_{n=1}^\infty$ converges on $\mu$ weakly, where $\{\mu_n\}_{...
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Convergence of $\int_0^1 \int_0^1 (1-xy)^{-a}dxdy$
How can I investigate the convergence of $\int_0^1 \int_0^1 (1-xy)^{-a}dxdy$ where $a \in (0, \infty)$? I know that one approach is to explicitly find the antiderivative but that seemed too laborious. ...
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Showing that the Dirac comb is a tempered distribution
Context: I am currently working through Chapter $8$ of Anders Vretblad's Fourier Analysis and Its Applications. This particular chapter focuses on distributions, and builds up to the Fourier transform ...
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Why doesn't the Riemann Zeta Function have zeroes at positive even integers?
According to the Riemann Functional Equation (source: https://en.wikipedia.org/wiki/Riemann_zeta_function) the Zeta Function is equal to itself multiplied by a bunch of stuff, including $$sin(πs/2)$$ ...
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How to solve a system of nonlinear equations with inequality constraints and ensure convergence?
Now I have the nonlinear equations with inequality constraints as below:
\begin{equation}
\label{equ:kkt}
\begin{aligned}
\left\{
\begin{aligned}
&\sum_{i=1}^M a_i \exp \left( -\frac{u_i}{\tau} ...
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How to calculate convergence rate of gradient descent
I am researching on gradient descent. I am looking at the convex case with Lipschitz-continous gradients.
For that I'm using Nesterov's "Lectures on convex optimitzation".
His result for the ...
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If $A\subset \mathbb{N}$ is large, then does $\sum_{n\in A} \frac{\vert \sin n \vert }{n}$ diverge also?
If $A\subset \mathbb{N}$ is large, that is, $\displaystyle\sum_{n\in A}
\frac{1}{n}$ diverges, then does $\displaystyle\sum_{n\in A}
\frac{\vert \sin n \vert }{n}$ diverge also?
I know that $\...
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Raabe test reference request
I just discovered that there is a convergence test called Raabe test
If $\displaystyle \lim_{n \to \infty}\inf \left\{ n \left(\frac{a_n}{a_{n+1}}-1 \right) \right\} >1 $ then the series $ \sum_{...
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If $\sum\frac{1}{a_n}$ diverges, then $\sum\frac{1}{n\max_{1\leq k\leq n}\frac{a_{n+1} - a_k}{n-k+1}}$ diverges.
Let $(a_n)$ be a strictly increasing sequence of positive real numbers, and denote
$\Delta a_n:= a_{n+1} - a_n.$
We know that if $\displaystyle\sum_{n\in \mathbb{N}} \frac{1}{a_n}$ diverges, then $\...
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Can someone solve for this series [closed]
$t_2= \sqrt{\sqrt 2+1}$
$t_4=\sqrt{\sqrt{\sqrt{\sqrt4+3}+2}+1}$
And $t_n$ follows the same pattern then does it converge? And if yes then to what value?
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Convergence Analysis of Symmetric Gauss Seidel Method for strictly row diagonally dominant matrices [closed]
I am trying to show that the Symmetric Gauss Seidel Method converges for SRDD matrices but am having trouble. I know I need to show that the spectral radius of the iteration matrix is less than 1. For ...
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This Special sequence convergent or divergent [closed]
If a(n+1)=a(n)^2-n-1 , does the sequence converge to 0 for any value of a(1)? Here a(n+1) denotes n+1 th term and a(n) denotes n th term.
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Can a nonzero complex power series have an uncountable set of complex roots?
Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing ...
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Convergence of sequence including binomial coefficients
While investigating the risk consistency of a particular machine-learning algorithm, I encountered the following convergence problem.
I would appreciate it if someone could point me in the right ...
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Convergence of a certain infinite product [closed]
Let $N$ and $k$ be a positive integers. I believe that the product
$$
\left( \frac{N-1}{N}\right) \cdot \left(\frac{N+k-1}{N+k} \right) \cdot \left( \frac{N+2k-1}{N+2k} \right) \cdot \left( \frac{N+...
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Order of Convergence in Simulations [closed]
I have made a simulation of Buffon's Needle Problem and I am trying to show that it converges, however, when I try estimating the order of convergence using Euler's method I get different answers each ...
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Why does this sum not diverge?
So I'm asking this question in general but with a motivating example:
Say we have a function $$f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
So both the top ($x^n$) and bottom ($n!$) of this function grow -...
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Convergence and sum of the series $\sum\left(L-a_n\right)^2$, where $a_0=0$ and $a_{n+1}=(a_n)^2+\frac{1}{4}$ and $L=\lim\limits_{n\to\infty} a_n$
I am trying to find out whether the series $\sum\left(L-a_n\right)^2$ converges (where $\{a_n\}$ is the sequence defined by the recurrence relation $a_0=0$ and $a_{n+1}=(a_n)^2+\frac{1}{4}$ and $L$ is ...
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Convergence of the Infinite Series of Reciprocals of Primes to 1
I am a junior high school student in Japan and I came up with an idea for something, but I don't know the name of it at all, even though I looked it up. Furthermore, I know only a little about high ...
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Show convergence of a sequence using $\varepsilon$-definition
I would like to show that the sequence $\frac{4n^2+3n}{n^2-4n+4}$ converges to $4$ as $n \to \infty$ by using the $\varepsilon$-definition of convergence. I am aware that it would probably be easier ...
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Find convergence a.s and in probability of $X_n$ if $P(X_n = 0) = \frac{1}{2n^\lambda}$ and $P(X_n = 2) = 1 - P(X_n=0)$
Given $\lambda > 0$ I want to find the convergence a.s and the convergence in probability of $X_n$ if $P(X_n = 0) = \frac{1}{2n^\lambda}$ and $P(X_n = 2) = 1 - P(X_n=0)$.
For the convergence a.s my ...
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Given $(U_n)_n$ a sequence of i.i.d random variables $U[0,1]$, find convergence of $1_{(0,1/n)}(U_1)$ and $1_{(0,1/n)}(U_n)$
Given $(U_n)_n$ a sequence of i.i.d random variables $U[0,1]$. I want to find the convergence in probability and a.s of the sequences $X_n = 1_{(0,1/n)}(U_1)$ and $Y_n = 1_{(0,1/n)}(U_n)$.
I don't ...
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Is there a convergent improper integral for an unbounded function?
I've been wondering if an improper integral (e.g. upper bound is $\infty$) over a function that is unbounded on the integration interval can converge.
Specifically, I know the integral $\int_1^\infty \...
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Convergence Test for Series $\displaystyle\sum_{n=1}^{\infty} \frac{n^{n}}{(n+1)^{n+1}}$
I am trying to determine the convergence of the series:
$$
\sum_{n=1}^{\infty} \frac{n^{n}}{(n+1)^{n+1}} = 1 + \frac{1^1}{2^2} + \frac{2^2}{3^3} + \frac{3^3}{4^4} + \frac{4^4}{5^5} + \dots \text{(to ...
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Finding Newton method order of convergence
I'm trying to determine how you find the order of convergence of newton's method. I have the formula $$\frac {|x^*-x_{n+1}|}{ |x^*-x_n|^q} = \alpha$$ I'm setting $q=2$ to test for quadratic ...
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Check the convergence of the series $\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$ and $\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$?
Check the convergence of the following series
$$\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$$ and $$\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$$
My attempt:
I tried Ratio test. I got
\begin{align}...
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convergence of scalar product of vectors without convergence of vectors
Let $(v,v_0,v_1,\dots)$ and $(w,w_0,w_1,\dots)$ be two infinite sequences of vectors of the closed unit ball of $\mathbb{R}^d$ (for the euclidean norm $\left\|.\right\|$, $d > 1$).
When $\left\|v_n-...
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If $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$, show that the sequence converges and find the limit.
If $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$, $\forall n\in N$, then the sequence converges, so find its limit.
So I list the elements of the sequence as follows:
$$1, 2, \frac{5}{2}, \frac{13}{5}, \frac{...
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Convergence of the series $\sum\limits_{n=1}^{\infty} \frac{(n!)^3 x^n}{n(3n)!}$
I am studying the convergence of the following series: $\sum\limits_{n=1}^{\infty} \frac{(n!)^3 x^n}{n(3n)!}$.
I have proceeded as follows:
$$\begin{align*}
\left|\frac{\frac{(n+1)!^3x^{n+1}}{(n+1)(3(...
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Is the given sequence cauchy , convergent
Question: Let $\\{a_n\\}$ be a sequence that satisfies
$|a_n| < 2$ ,
$|a_{n+2}-a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2-a_{n}^2|$.
Then
a) $\\{a_n\\}$ is a cauchy sequence
b) $\\{a_n\\}$ is a bounded ...
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Domain of convergence for a power series
I have a homework question that we have not discussed at all how to do in class, so any help or explanations would be greatly appreciated! I don't know where to start.
In each of the following cases, ...
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Are the sequences of families for summability or improper summability:
I would like to analyse the following sequences (a, b and c) of families for summability or improper summability:
a) $ a_{i}:=i^{-\alpha} $ for $ \alpha>0 $ :
b) $a_{i}:=(-1)^{i} i^{-\alpha}$ for $\...
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How to find the summation of the above trigonometric series without using desmos?
How to find the summation of
$$
\sin(2 + \sin(2 + \sin(2 + \cdots \infty)))?
$$
I am trying this question by denoting the above summation as $S$. Therefore,
$$
S = \sin(2 + \sin(2 + \sin(2 + \cdots \...
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Show that the function converges at a = 0
Show that the function $ f:(−1,0)∪(0,1) → R , x → { 1 \over {1 \over x} +1} $ converges at $ a = 0$
The function ${ 1 \over {1 \over x} +1}$ is defined on the interval $ f:(−1,0)∪(0,1) $ excluding ...
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Does $\sum_{n=1}^\infty \frac{1}{n \lfloor f(n) \rfloor+(\lfloor f(n+1) \rfloor-\lfloor f(n) \rfloor)n^2}$ converge?
Does
$$\sum_{n=1}^\infty \frac{1}{n \lfloor f(n) \rfloor+(\lfloor f(n+1) \rfloor-\lfloor f(n) \rfloor)n^2}$$ converge for all non-constant, positive, increasing function $f$ that grows sublinearly?
...
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Problem on convergence of infinite product
Question : The infinite product
$\prod_{n=2}^\infty(1-\frac{2}{n(n+1)})$
converges to what value ?
My attempt : I tried using partial fractions. The above product became
$\prod_{n=2}^\infty(1-\frac{2}{...