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Questions tagged [conditional-convergence]

This tag is for questions related to conditional convergence. A series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

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Convergence of Riemann zeta function [duplicate]

I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
Leif Sabellek's user avatar
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1 answer
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Conditional convergent series implies existence of rearrangement that diverges: Doesn't the sum of the negative terms tend to $-\infty$?

In the proof for the Riemann Series Theorem that I'm reading, the author is currently establishing the existence of a divergent rearrangement of an infinite series given that the original series ...
lightweaver's user avatar
2 votes
1 answer
140 views

how do you compute the value of $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$

I know that the series $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging. I try to evaluate out ...
ToThichToan's user avatar
2 votes
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42 views

Rearranging conditionally convergent series without changing the limit

Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that the series $\sum a_n$ is conditionally convergent, i.e. the limit $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
Jonas's user avatar
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3 votes
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To decompose a conditionally convergent series into a partial bounded series and another decreasing series

In the first-year mathematics analysis course, the instructor assigned a problem on the convergence of series. We are given that a series $\sum_{n=1}^\infty A_n$ converges absolutely if $\sum_{n=1}^\...
Liping Li's user avatar
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1 answer
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Does $a_n$ converge if $a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$? [closed]

Does $a_n$ converge if$a_{n+1} = a_n + \frac{1}{e^{a_n}+1}$? Or does its convergence depend on $a_0$? As described in the title, it seems intuitively that it should converge, but I don't know how to ...
n yk's user avatar
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1 answer
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Find the conditional expectation $E[X \mid X \leq p]$

Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X \...
Sumit Singh's user avatar
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0 answers
25 views

Convergence of Alternating Series Involving Cosine Term and Square Root

I am working on a series and attempting to determine its conditional convergence using the alternating series test. $$ \sum_{n=1}^{\infty} \frac{\cos\left(\frac{\pi}{4} + 2\pi n\right)}{\sqrt{n}} $$ I ...
Sai Charan Petchetti's user avatar
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1 answer
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For conditionally convergent series $\sum a_n,\exists (m_n)_{n\in\mathbb{N}}$ with $(n-1)k<m_n\leq n k,$ s.t. $\sum_{n\in\mathbb{N}} a_{m_n}=\alpha.$

Properties of conditionally convergent series $\ \displaystyle\sum a_n\ $: The sum of the positive terms is $+\infty;\ $ the sum of the negative terms is $-\infty.$ $\displaystyle\lim_{\substack{ { n\...
Adam Rubinson's user avatar
3 votes
1 answer
55 views

Re-ordering conditionally convergent series over $\Bbb C$.

For a conditionally convergent series $\sum a_n$ over $\Bbb C$, let $M\subset \Bbb N$ be a set such that the sub-series $\displaystyle \sum_{n\in M} a_n$ converges absolutely. $\sigma:\Bbb N \to \...
emacs drives me nuts's user avatar
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1 answer
155 views

What happens to EX if E|X| is infinity?

---------original question---------------- According to my professor, we can divide $X$ into $X_+=\max(X,0)$ and $X_-=-\min(X,0)$, both nonnegative. And we have $EX=EX_+-EX_-$ and $E|X|=EX_++EX_-$ For ...
Xiangyu Cui's user avatar
3 votes
2 answers
274 views

Can a real function have convergence that oscillates depending on its derivative?

I recently read a post on here in which a user asked if there existed a function, $$\lim_{x\rightarrow\infty}f(x)$$ is convergent, but; $$\lim_{x\rightarrow\infty}f'(x)$$ does not converge. I wanted ...
Amy Skinner's user avatar
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0 answers
24 views

Does Reordering and Regrouping the functions in a Conditional Infinite Series (as described) terms change its Sum?

Consider any infinite conditional convergent series of the form: $Sum_0=\sum_{n=1}^\infty f(n) + g(n) + h(n) = C_0$. Here each 'term' consists of 3 separate functions $f(n), g(n), h(n)$. Now assume ...
stack.tarandeep's user avatar
1 vote
1 answer
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How to prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent?

Let $f_n(x)=x^n+nx-1$, let $a_n$ denote its unique positive root. Then prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent. Below is my solution. First, $$a_n(a_n^{n-1}+n)=1,$$ because $...
Ychen's user avatar
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3 votes
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Is this proof that $\sum\cos(n)\frac{(n+1)^n}{n^{n+1}}$ converges conditionally, correct?

Determine if the following series converges absolutely, converges conditionally or diverges: $$\sum\cos(n)\frac{(n+1)^n}{n^{n+1}}.$$ The series $\sum\cos n $ is bounded and the sequence $a_n=1/n$ is ...
Hilbert's user avatar
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1 answer
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Convergence of a series given that the series with a specific parentheses insertion converges

I am given a sequence $(a_n)_n$ such that $|a_n|\le\frac{1}{n}$ and the sequence $b_k=\sum_{n=k^2}^{(k+1)^2-1}a_n$ and I am given that the series $\sum_k b_k$ converges. I need to show that the ...
Ofek Aman's user avatar
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0 answers
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On Convergence of Alternating Harmonic Series

I am almost new with Mathematical Analysis and I see something that made me to think! It is proved that Alternating Harmonic Series is convergent to ln(2). What if ...
Mehdi Mowlavi's user avatar
2 votes
0 answers
59 views

Necessary and Sufficient Condition for A Particular Sum Rearrangement

Let $\phi:\mathbb{N}\rightarrow\mathbb{N}$ be a rearrangement of $\mathbb{N}$ (a bijection). I am searching for a condition equivalent to: $$$$ For all complex sequences $(\alpha_n)$, there exists ...
Miles Gould's user avatar
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how does rearranging terms of harmonic series result in different values?

In the book In Pursuit of Zeta-3 the author states that rearranging the terms of a harmonic series results in a different sum. But isn't addition commutative? How ...
user1078's user avatar
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Conditionally convergent series value

Find all values of $p$ such that the series $\displaystyle\sum_{j=1}^{\infty}\dfrac{(-1)^j\cdot\log(j)}{j^p}$ is conditionally convergent. I know that the alternating series test states that if a ...
user1197542's user avatar
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0 answers
58 views

$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent arrangement prime number theorem related?

$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent neighbor prime number theorem related? $ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\...
Marvin Ray Burns's user avatar
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0 answers
28 views

Number of permutations for a conditional convergence series

I'm asking the next thing: If we have a rational conditionally convergent series: $$\beta=\sum_{i=0}^\infty q_i \; \; \; ({q_i}\in\Bbb Q)$$ Then we know thanks to Riemann that it may be rearranged to ...
tomascatuxo's user avatar
4 votes
1 answer
90 views

Finding a conditionally convergent series of functions in $C[0,1]$ with supremum norm w.r.t Faber-Schauder system.

Let, \begin{align*} \alpha(x) &= 1 \\ \beta(x) &= x\\ s_{n,k}(x) &= \max\{1-|2(2^nx-k)-1|, 0\} \text{ for } 0 \le n \text{ and } 0 \le k \le 2^{n}-1 \end{...
Klomanad's user avatar
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0 answers
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Absolutely/Conditional Convergence [duplicate]

Is there an example of a power series that is conditionally convergent at one endpoint and absolutely cinvergent at another? Do such series exist at all?
Elnur Khalilov's user avatar
2 votes
1 answer
102 views

Convergence of alternating series $\sum_{n=2}^{\infty}\frac{(-1)^n}{\ln^2(n) \sqrt[n]{n!}}$

Convergence of alternating series $\displaystyle\sum_{n=2}^{\infty}\frac{(-1)^n}{\ln^2(n) \sqrt[n]{n!}}$ I think it is clear that the series at least conditionally converges by the alternating series ...
Sungwon's user avatar
  • 23
0 votes
2 answers
68 views

I think $\sum_{\alpha\in A}x_\alpha=\sum_{\beta\in B}x_{\phi(\beta)}$ always holds even if $(x_\alpha)_{\alpha\in A}$ is not absolutely convergent.

The following is from "An introduction to measure theory" by Terence Tao. Motivated by this, given any collection $(x_\alpha)_{\alpha\in A}$ of numbers $x_\alpha\in [0,+\infty]$ indexed by ...
tchappy ha's user avatar
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0 votes
1 answer
165 views

Does the alternating p-series converge conditionally for $p\leq0$?

I'm aware that the alternating p-series $\sum_{n=1}^{\infty}(-1)^n1/n^p$ converges absolutely for $p>1$ and conditionally for $0<p\leq1$, and these are fairly straightforward to prove. But what ...
pileafilea's user avatar
1 vote
0 answers
29 views

Terminology for the "correct" value of a conditionally convergent sum

As far as I understood, the terms in a conditionally convergent series can be rearranged so that the sum converges to any value at all. I was reading the book 'Lattice Sums, Then and Now' by Borwein ...
Tom's user avatar
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0 votes
1 answer
59 views

Series with $\lim \sup$ and $\lim \inf$ being $+$ and $-\infty$

Do you have an example of a series whose partial sums' $\lim \sup = \infty$ and whose partial sums' $\lim \inf = -\infty$ ? I feel like it's like repeatingly adding a whole bunch of positive terms ...
niobium's user avatar
  • 1,221
2 votes
1 answer
53 views

How to prove this characterisation of convergence by distribution?

We solved the following problem in class and I do not understand what happens. Definition: A sequence of random variables $W_1, W_2, \ldots$ converges in distribution to random variable $W$ if for ...
Jesus's user avatar
  • 1,798
2 votes
1 answer
88 views

Madelung's constant by neutral spheres

Madelung's Constant is informally defined as: $$M = \sum_{i,j,k=-\infty}^{+\infty} \frac{(-1)^{i + j + k}}{\sqrt{i^2 + j^2 + k^2}}$$ where the sum does not include the $i = j = k = 0$ term. The sum is ...
QCD_IS_GOOD's user avatar
  • 2,327
0 votes
1 answer
15 views

How to get the last function in the picture by o.1 and o.2 equations? (Conditional distribution)

[for the last equation, is it shows that the sum of P(x given y) times p(y)? And i do not sure how this equation form by o.1 and o.2, please give me a help. Best wishes 1,
Sophie Ma's user avatar
1 vote
0 answers
47 views

Example for convergence and absolute convergence abscissa

Let $f\in L^1_{loc}(\mathbb{R})$ a locally integrable function, with $f:\mathbb{R}\rightarrow\mathbb{C}$. We say that $\int_0^{\infty} f(t) dt$ converges if the limit \begin{equation} \lim_{b\to\infty}...
JaviLark01's user avatar
1 vote
1 answer
95 views

Prove convergence of alternating sum with positive and negative terms

I am trying to find under which conditions does the sum below converge to 0, $\sum_{n=1}^{\infty}{(-1)}^{n+1}\cos(b\log(n))({n}^{a-1}-{n}^{-a})=0,$ with b and a real numbers, with 0 < a < 1. I ...
stanisverylow's user avatar
4 votes
1 answer
72 views

Given a conditionally convergent series, can we group terms together retaining order, so that the quotient of consecutive groups tends to $1$?

Suppose $\displaystyle\sum_n a_n$ is a conditionally convergent series, that is, $\displaystyle\sum_n a_n$ converges and $\displaystyle\sum_n \vert a_n \vert = +\infty.$ Proposition: There exists $\ ...
Adam Rubinson's user avatar
0 votes
0 answers
52 views

Conditional uniform convergence in probability

I am wondering if it makes sense to say that a (uniform) convergence in probability is conditional on a random variable. Or maybe, there is another way to express that. Assume two real-valued random ...
Ari.stat's user avatar
  • 453
5 votes
1 answer
103 views

For which periodic sequences $(a_n)$ does the series $\sum \frac{a_n}n$ converge?

Let $(a_n)$ be some sequence of real (or maybe even complex) numbers. For which sequences does $$S=\sum_{n=1}^\infty \frac{a_n}n$$ converge to a finite value? Let $p$ denote the period of $a_n$, i.e. $...
emacs drives me nuts's user avatar
2 votes
0 answers
110 views

A power series that converges conditionally for all points on the radius of convergence?

For $a_n\in\Bbb C$ let $$f(z) = \sum_{k=0}^\infty a_n z^n \tag 1$$ be a power series with radius of convergence of 1, and $a_n$ such that the series converges for all $z\in\Bbb C$ with $|z|=1$. What's ...
emacs drives me nuts's user avatar
10 votes
0 answers
341 views

Can the Cauchy product of a conditionally convergent series with itself be absolutely convergent?

If $\sum_{n\ge 0} a_n$ and $\sum_{n\ge 0} b_n$ are two series, their Cauchy product is defined as $\sum_{n\ge 0} c_n$, where $c_n = \sum^n_{k=0} a_k b_{n-k}$. As this question points out, finding two ...
Jianing Song's user avatar
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3 votes
0 answers
148 views

Convergence of iterative map

I have the following iterative mapping: $$x_{n+1} = (N-n)^{-1} \frac{x_n}{f(x_n)} \left(C - \sum_{i=1}^n f(x_i)\right)$$ defined for $n \leq N$ and where $C > 0$ is some constant. I am trying to ...
Doc Stories's user avatar
13 votes
0 answers
194 views

Rearranging series and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. Let $...
Noah Schweber's user avatar
2 votes
0 answers
20 views

A problem in convergence and limit

For any non-negative integer $n$ and some finite $r$, we introduce the notation $n_k$ which indicates the number of $\{X_1, X_2, \cdots , X_n\}$ belonging to the $k$-th distribution type, for $k=1, 2, ...
Statistics 's user avatar
0 votes
0 answers
83 views

A new convergence problem for the conditional expectation

You have risks $X_1$, $X_2$, ... (they are assumed to be independent, but not necessarily identically distributed) and $S_n= X_1 + X_2 + \cdots +X_n$ QUESTION: under what reasonable conditions we have ...
Statistics 's user avatar
0 votes
0 answers
168 views

Conditional convergence of the Hadamard product

The product $$\prod_\rho \left(1-\frac{s}{\rho}\right)$$ where $\rho$ ranges over all zeros of the Riemann zeta function is often referred to as the Hadamard product (because its convergence was first ...
Valerio's user avatar
  • 370
2 votes
1 answer
80 views

If we can't change the order of the terms of a conditionally convergent series, but can change the sign of terms, can we get whatever number we want?

Let $\ \displaystyle\sum_{n\in\mathbb{N}} x_n\ $ be a conditionally convergent series, and we can choose $\ a_j=-1\ $ or $\ a_j=1\ $ for each $j\in\mathbb{N}.$ Let $\ \beta\in\mathbb{R}.$ Can we ...
Adam Rubinson's user avatar
0 votes
1 answer
140 views

Find out constants$~a,b,c,d~$such that$~\lim_{x\to0}\frac{\sin^{}\left(3x\right)-\left(ax^{2}+bx+c\right)}{x^{3}}=d~$is satisfied

$$\left(a,b,c,d:=\text{constants}\right)~~\wedge~~\left(d\neq0\right)$$ I want to find out the formula(s)or value(s)of the above constants which satisfy the following equation. $$\lim_{x\to0}\frac{\...
electrical apprentice's user avatar
2 votes
1 answer
659 views

Integral representation of Bessel function $J_1(x)$

In "The Handbook of Mathematical Functions" by Abramovitz and Stegun, according to Eq. 9.1.24, \begin{align} J_0(x)=&\frac{2}{\pi}\int_{1}^\infty \frac{\sin(xt)}{\sqrt{t^2-1}}dt,\quad x&...
evening silver fox's user avatar
2 votes
1 answer
108 views

Determine if the series converges absolutely, converges conditionally, or diverges.

Determine if the series converges absolutely, converges conditionally, or diverges. Find the exact value for the sum of the convergent series. $$ 1-\frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5}...
Todd Jones's user avatar
0 votes
1 answer
545 views

Determine if the series converges absolutely, conditionally, or diverges.

Determine if the series converges absolutely, converges conditionally, or diverges. Find the exact value for the sum of the convergent series. $$1-\frac{1}{5} - \frac{1}{5^2} + \frac{1}{5^3} - \frac{1}...
Todd Jones's user avatar
2 votes
0 answers
60 views

If $a_n \to a$ and $b_n \to b$, then $\sum_{k=1}^n \frac{a_kb_{n-k}}{n} \to ab$ [duplicate]

If $a_n \to a$ and $b_n \to b$, then $\sum_{k=1}^n \frac{a_kb_{n-k}}{n} \to ab$. Is this an application of Cesàro series? If the term $\sum_{k=1}^n {a_kb_{n-k}}$ is a partial sum of some sequence that ...
jerry guna's user avatar

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