Questions tagged [convergence-divergence]
Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.
22,521 questions
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Divergence Test [closed]
The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...
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Is $|\sin(n) | \le 1/n^2$ for infinitely many natural numbers $n$?
I have already finished my analysis classes with good grades. But lately I am playing with Geogebra and wonder if there exists a sequence of natural numbers $(n_k)$ such that $$|\sin{n_k}|\leq\frac{1}{...
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1
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How to show that the ratio test for $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$ is unity?
According to Wolfram Alpha the ratio test equals $1$ for the series, $\sum_{n=1}^\infty\frac{(-1)^n\sqrt{n^2+1}}{n\ln n}$, hence convergence/divergence is inconclusive. The purpose of this question is ...
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6
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Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
I have this limit
$$\lim_{n\to \infty} \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$$ and and I tried to evaluate it in the following way:
First I set $x=\frac{k}{n^2}$ to make the expression ...
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7
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How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
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Convergence of $\sum_{n=1}^{\infty} a_n,$ $\text{ where }$ $a_n = \prod_{k=1}^{n} \sin^2(2^k x)$ $\text{ and }$ $x \in (-\infty, +\infty)$ [closed]
To determine the convergence of the series $$\sum_{n=1}^{\infty} a_n, \text{ where } a_n = \sin^2 x \sin^2 2x \dots \sin^2 2^{n}x \text{ and } x \in (-\infty, +\infty).$$
I attempted to use the ratio ...
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Why do we need \rho (utilization)<1 in queuing theory?
I’m studying basic queueing theory, in particular a single–server queue with one arrival stream and one server (a G/G/1 type setup).
Let
A be the interarrival time with mean 𝔼(A),
B be the service ...
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3
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2
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Martingales convergence a.s.
Suppose we know that
$P(X=1)=p, P(X=-1)=q=1-p$
We have a martingales
a) $M_n = (\frac{q}{p})^{S_n}$,
b) $M_n = S_n - n(p-q)$
where $S_n =\sum_{i=0}^n X_i$,( $X_i$ iid). How to show (in a simple way) ...
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Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
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Distribution of sum of product [closed]
Let $x_n\overset{p}{\to}c$ and $x_n\overset{d}{\to}N(0,\sigma^2)$ denote convergence in probability to a constant $c$ and convergence in distribution to a random normal variable (with some abuse of ...
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Determine the convergence or divergence of $\sum_{n=1}^\infty\frac{1}{n^{\arctan n}}$
There was a problem in my textbook:
Determine the convergence or divergence of $$
\sum_{n=1}^\infty\frac{1}{n^{\arctan n}}
$$
The method my instructor taught me is that, notice that this sum is ...
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Series that is known to converge/diverge but for which all these standard tests are inconclusive .
I have noticed that nearly every series I have been asked to analyze its convergence or divergence can be handled by the usual collection of tests: the limit test, Cauchy condensation, the integral ...
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If for every sequence $(x_n)\subset X-\{a\}$ with $\lim_{n\to\infty}x_n=a$, the sequence $(f(x_n))$ is convergent, then $\lim_{x\to a}f(x)$ exists?
I am studying the topic of limits of functions in $\mathbb{R}$ and I want to prove the following
Let $f:X\to \mathbb{R}$ with $X\subset \mathbb{R}$ and $a\in X'$. If for every sequence of points $x_n\...
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Is "the nth logarithm test" for series' convergence reliable?
Is it possible to determine the convergence of a series $\sum a_n$ by evaluating $$\lim_{n \to \infty} \log_n(a_n)$$ and comparing the result to $-1$? Specifically, if this limit is less than $-1$, ...
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2
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92
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Long term behaviour of the sequence $a_{n+1}=a_0^{a_n}$ for $a_0\in\mathbb{C}$
I am interested in the long term behaviour of the sequence $a_{n+1}=a_0^{a_n}$ where $a_0\in\mathbb{C}$. I am aware there are many sources discussing convergence of this sequence(for example) but I am ...
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Result of an absolutely convergent series?
I encountered this doubt when I began to study absolutely convergent series in $\mathbb{R}$.
If $\sum_{n=1}^{\infty}b_n$ is an absolutely convergent series and $\left(\frac{a_n}{b_n}\right)_{n\in\...
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1
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Are there different ROCs for $\mathcal{L}^{-1}(\frac{1}{s^2})$
To calculate inverse Laplace transform of $\frac{1}{s^2}$ I use integration:
$\int t e^{st} dt = t\frac{e^{-st}}{-s} - \int \frac{e^{-st}}{-s}dt.$
Here I clearly see that I need to put conditions $s$ ...
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154
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Struggling to show how to sum a specific series
I am trying to show that
$$\sum \left( \dfrac{1}{8} + \dfrac{2}{32} + \dfrac{5}{128} + \dfrac{14}{256} + \dfrac{28}{1024} + \dfrac{76}{2048} + \dfrac{151}{8192} + \dots \right) =1/2.$$
The ...
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Does every sequential convergence define a "nice" topology? [duplicate]
There have been a lot of discussions about this topic but I am still confused.
Here is the context: I have a set $E$ that I would like to endow with a topology $\mathcal{T}$.
Let us say that I have a ...
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145
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Question about closed form/ Convergence of $\int_{0}^{\infty}\sin\left(x\right)\left(\sin\arctan^{2}x-\sin\frac{\pi^{2}}{4}\right)dx$
I'm interested in the improper integral
$$
I = \int_{0}^{\infty}\sin(x)\Big(\sin(\arctan^{2}(x)) - \sin\!\Big(\frac{\pi^{2}}{4}\Big)\Big)\,dx.
$$
$
\textbf{Questions:}$
Does this integral converge (...
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Does this series $\sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(\frac{-n}{2})}$ converge? [closed]
In the series below, I'm aware that dividing a number by infinity yields zero, and I'm curious why Mathematica can't solve it.
$$
\sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(\frac{-n}{2})}
$$
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Convergence of a Zeta-Zeta function
Define $Z$ as a sort of zeta function defined over the powers of the non-trivial zeros $\rho$ of the Riemann zeta function (a meta-zeta function). So we have, $$Z(s) = \sum_{\rho} \frac{1}{\rho^s}$$ ...
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Associativity of the sum in an analytic function
Here are two definitions.
ANALYTIC MAP
Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if
$$
f(x)=\sum_{k=0}^{\infty}a_k(x)
$$
where
for each $ k$, the map $a_k \...
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Prove that $\sum a_n $ is divergent if $a_n>0,a_n^{\frac{1}{n}}+a_n-1=0$
Prove that $\sum a_n $ is divergent if $a_n>0,a_n^{\frac{1}{n}}+a_n-1=0$
We define $a_n>0,a_n^{\frac{1}{n}}+a_n-1=0$
. And my problem is to prove that $\sum a_n $ is divergent.I’ve tried to ...
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Real Analysis - Sequences - Basic Q
I'm starting to read Real Analysis. Consider below:
$$
\lim_{n\to\infty} \sum_{i=1}^{n}x_i = \sum_{i=1}^{\infty}x_i
$$
This seems obvious and intuitive, but:
Q1 - How do you prove this (rigorously, at ...
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What am I doing wrong with this Taylor Series expansion
I am trying to find the Taylor Series expansion for sin(x) at $\frac{π}{2}$.
I found the series and confirmed it with the key, however when I graph it my solution seems to be shifted down 1 and adding ...
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Is this convergence criterion theorem for improper integrals, obtained by analogy with d'Alembert's ratio test for series, correct? How to prove?
The origin of the question: Recently, after finishing the study of improper integrals on infinite interval for single-variable functions, I began learning about series. The professor reminded us to ...
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SGD convergence when visit basin of attraction infinitely often
Consider a discrete stochastic system with components $(x_k, y_k)$ updated as follows.
If all components are strictly positive, i.e. $x_k > 0$, $y_k > 0$, then
\begin{aligned}
x_{k+1} &= x_k ...
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Convergence Conditions for Three Families of Infinite Series
Consider three types of sums:$$R=\sum_{i=0}^\infty\frac{1}{p^{i+q}}\\S=\sum_{i=0}^\infty\frac{1}{(i+p)^q}\\T=\sum_{i=0}^\infty\frac{1}{(i+p)^{i+q}}$$in which $p,q\in\Bbb{R}$.
How will the astringency ...
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Definition of convergence and stopping time
I have trouble understanding whether the usage of stopping time is correct. Suppose I have a sequence $\{a_t\}_t$ and $\lim_t a_t = c$ almost surely (a.s), where $|c| < \infty$. By definition of ...
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Is there a natural def of divergence on posets
Similar to how we think of divergence of $\mathbb{N}$ or $\mathbb{R}$ one can naturally inherit this concept to talk about divergence on those sets as with the std order. But what about in a more ...
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Order of convergence of Newton Raphson Method (modified)
I understand the definition of the order of convergence: if $(x_n)$ is a sequence that converges to a root $r$, its order of convergence is the largest positive constant $\alpha$ such that
$$
\lim_{n\...
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Does the series $\displaystyle \sum_{n=1}^m (-1)^n n^{\sigma-1} \cos(t\ln(n)) \sum_{k=1}^n (-1)^k k^{\sigma-1} \cos(t\ln(k))$ diverge?
For $s=\sigma +it ~$ ($0<\sigma<1, t\in \mathbb{R}$), under the hypothesis that $\displaystyle ~\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = 0$, I was wondering if the following series,...
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Understanding the proof of Dirichlet's test for convergence - part 2.
In the previous part to this question
I have the following proof of Dirichlet's convergence test:
Theorem (Dirichlet’s Test).
If the sequence $\{a_n\}$ is monotonically decreasing with $a_n \to 0$ as ...
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1
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Understanding the proof of Dirichlet's test for convergence - part 1.
I have the following proof of Dirichlet's convergence test:
Theorem (Dirichlet’s Test).
If the sequence $\{a_n\}$ is monotonically decreasing with $a_n \to 0$ as $n \to \infty$, and if the partial ...
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A question related to a convergence of Levy measures
I am working on the preservation of symmetry for Lévy measures under a specific type of convergence. First, recall that a Lévy measure $\mu$ on $\mathbb{R}^k$ is symmetric if $\mu(E) = \mu(-E)$ for ...
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If $X_n \to X$ in $L^r$ for some positive integer $r$, then $E[X_n^r] \to E[X^r]$.
I want to prove: If $X_n \to X$ in $L^r$ (i.e., $E[|X_n - X|^r] \to 0$) for some positive integer $r$, then $E[X_n^r] \to E[X^r]$.
I tried using Minkowski's inequality: $\left| \|X_n\|_r - \|X\|_r \...
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Is there a reliable reference to the proof of the statement, 'A bounded monotone series converges' (not sequences)?
I'm trying to find the name of the theorem, or a reference stating something like: 'If a series is bounded and monotone, then it converges.'
I know the statement is true for sequences, as I have the ...
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$\int f = \liminf \int f_n \implies \int f = \lim \int f_n$ [duplicate]
Suppose $|f_n| \le g \in L^1$ and $f_n \to f$ in measure. Then $\int f = \lim \int f_n$.
I managed to show slightly weaker equality that $\int f = \liminf \int f_n$, but I have no idea how to get ...
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Prove weighted mean of convergent sequence converges
Let $\lim\limits_{n\to\infty}a_n=a$ and $b_i>0$ for all positive integers $i$. Prove that
$$c_n=\frac{a_1b_1+a_2b_2+\cdots+a_nb_n}{b_1+b_2+\cdots+b_n}$$
is convergent.
Here is my proof. Since $\{...
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2
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Convergence of $\prod^{\infty}_{n=1}\left(\frac{2n}{2n-1}\right)$
Does $\displaystyle\prod^{\infty}_{n=1}\left(\frac{2n}{2n-1}\right)$ converge to a finite value?
If you take the natural logarithm:
\begin{equation*}
\ln\left(\prod^{\infty}_{n=1}\left(\frac{2n}{2n-1}\...
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1
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for $a_n >0$ $\forall$ $n \geq 1$, Does the condition $a_{n+1} \leq a_n + \frac{1}{(n+1)^2}$ imply convergence of $(a_n)$? [duplicate]
Problems in Real Analysis Ex 1.6.48
Let $(a_n)_{n \geq 1}$ be a sequence of positive real numbers such that for every $n \geq 1$
$$a_{n+1} \leq a_n + \frac{1}{(n+1)^2}.$$
Prove that the sequence $(...
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1
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Is this series really convergent?
According to the prime number theorem $\frac{x}{\ln x}$ is approximately the number of primes below x and gets better as x gets larger, from this we get the result that d(P) = 0 which means that the ...
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1
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409
views
Why does this discrete product built from floor and ceiling of squares converge to pi or 1?
I have been experimenting with a structure I call the Discrete Square Residual Structure (DSRS).
For a fixed integer $\mu > 0$, define
$U(n) = \lceil \tfrac{n^2}{\mu} \rceil, \quad L(n) = \lfloor \...
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1
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Need help understanding the proof of the 'ratio test for sequences' (not series)
I think the parts boxed in red for the following 'ratio test for sequences' are incorrect:
Theorem:
Let $x_{n}$ be a sequence of positive numbers and let
$$L = \lim_{n \to \infty} \frac{x_{n+1}}{x_{n}}...
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3
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$\lim n^{2}\left(\log{2} - \sum_{1\leq j\leq 2n-1}\frac{2}{2n + j}\right) = $?
Problem: Let
$$
a_{n} = \frac{2}{2n+1} + \frac{2}{2n+3} + \cdots \frac{2}{4n-1}.
$$
Then find the following limit:
$$
\lim_{n} n^{2}\left(\log{2} - a_{n}\right).
$$
My attempt: Note that
$$
\frac{2(...
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0
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Why is R-convergence a useful notion?
Consider the definition of $R$-convergence as given in Definition 9.2.1 of "Iterative solutions of nonlinear equations in several variables" by Ortega and Rheinbolt.
Let $A$ be a fixed-...
0
votes
1
answer
125
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Split a series in a sum of series
Suppose that $\sum_{n = 0}^\infty a_n$ is absolutely convergent and $\{P_1,\ldots, P_r\}$ is a partition of $\mathbb{N}$ (i.e. $\bigcup_{i=1}^r P_i = \mathbb{N}$ and $P_i \cap P_j = \emptyset$ for any ...
- 1,482
6
votes
0
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108
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Norm convergence in Lie-Trotter formula
I am studying the Lie-Trotter formula for operator exponentials. Let $H$ be a Hilbert space and $A$, $B$ be self-adjoint operators on $H$. The classical Lie-Trotter formula (see M. Reed, B. Simon. ...
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8
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2
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170
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Lower limit of the means of independent random variables
Let $X_1,X_2,\cdots$ be a series of independent random variables with
$$\mathbb{P}(X_n=-1)=\frac{1}{2},\quad \mathbb{P}(X_n=4n) = \frac{1}{8n},\quad \mathbb{P}(X_n=0)=\frac{1}{2}-\frac{1}{8n}$$
Let $...
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