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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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Improper Lebesgue integral $\int_{\mathbb R}\frac{\sin(x)}x~\mathrm dx$

While thinking over the idea of an improper Lebesgue integral, I came up with the following: $$\int_0^\infty\mu\{x~|~f(x)>t\}-\mu\{x~|~f(x)<-t\}~\mathrm dt$$ with $\mu$ being the Lebesgue ...
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30 views

Is square of Poisson kernel conditionally convergent?

Poisson kernel is defined for $\theta\in [0,2\pi)$ as $$P_r(\theta) = \sum\limits_{n\in\mathbb{Z}} r^{|n|}e^{in\theta}$$ which equals $$P_r(\theta) = \frac{1-r^2}{r^2-2r\cos (\theta) + 1 }.$$ For this ...
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3answers
44 views

Convergence of the series below

$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$ I did: $$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$ $$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$ So diverges by the Ratio ...
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3answers
32 views

The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0?

The power series $\sum \frac{(x-b)^n}{na^n}$ with a,b>0 ? How do i show for which x the series is conditionally convergent? Do i have to express in terms of a and b. or something like that?
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1answer
30 views

Could I get an explanation on why this would conditionally converge?

$$\sum\limits_{n=2}^\infty \frac{\cos(n\pi)}{\ln(n)^2}$$ I'm not sure how this would conditionally converge, according to my calculations I would assume it's absolutely converge.
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1answer
30 views

Numerical Solver Exercise

Sensitivity to initial conditions is well illustrated by a little target practice with your numerical solver. Enter $x' = x^2 - t$ into your numerical solver, and then experiment with initial ...
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3answers
32 views

Is $\sum_{n=1}^{\infty} \frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$ divergent, conditionally convergent, or absolutely convergent?

The first thing I did was create a sequence ($A$) for what was inside of the sum, then I created another sequence ($B$) that is related to $A$. $$A=\frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$$ $$B=\frac{1}{n^2}...
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2answers
79 views

Show that if $\sum a_n$ converges absolutely, then $\sum (\tfrac{n+1}{n})a_n$ also converges absolutely.

Prove that if $\sum_{n=1}^\infty a_n$ is convergent, then the series $$\sum_{n=1}^\infty \left(\frac{n+1}{n}\right)a_n$$ is also convergent. Edit: Originally, I thought about doing a limit comparison ...
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1answer
49 views

Conditional convergence of $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$

I am working on the exercise of determining when $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ converges conditionally. I know if $p\leq0$ it cannot converge by the vanishing condition, and if $p>1$ the ...
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17 views

Rearranging Terms of Conditionally Convergent Series

I know about Riemann’s Rearrangement Theorem. But certainly you can rearrange some of the terms of a conditional series and still arrive at the same series value. For example, if you only rearrange a ...
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2answers
42 views

Working out which x for the Conditional Convergence of a Series

I'v got roughly half way through this question: For (fixed) x which is an element of the real numbers, consider the series $\sum_{n=1}^\infty \frac{x^{n-1}}{2^nn} $ For which x does this series ...
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3answers
50 views

Why is $\sum_{k=2}^{\infty} \frac{(-1)^k}{k\ln k}$ conditionally convergent?

I was solving this question, and I thought it would be absolutely convergent because when you set the modulus, you would get $$\sum_{k=2}^{\infty} \frac{1}{k\ln k}.$$ But the answer says it is ...
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1answer
38 views

critical case in $L^p$ convergence

Let $f_n$ be a bounded sequence in $L^1\cap L^{p}$ with $1<p<\infty$. Assume that $f_n\to f$ strongly in $L^1$. Then basically we have $f_n\to f$ strongly in $L^q$ for all $1\leq q<p$. But we ...
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1answer
42 views

Good Book for methods of Convergence

First of all I'm interested in methods of convergence (for sums and integrals) so are there good books which tackle these methods one after the other and with examples that are not too trivial? For ...
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1answer
44 views

Absolute or conditional convergence?

Determine whether the series: $$\sum_{n=1}^\infty (-1)^n \frac {2n^2+3n+4} {2n^4 + 3}$$ converges absolutely, conditionally or diverges. I know the series converges conditionally using alternating ...
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3answers
143 views

determine series convergence.

Determine whether this series converges: $\sum_{n=1}^\infty \cos(n^2\pi) (\sqrt{n+11} -\sqrt{n+2}) $ I know that $\lim a_{n} = 0$ and that this series alternates because of $cos(n^2\pi)$, but don't ...
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1answer
84 views

How to use AND condition in Desmos

Sorry maybe it's not typical mathematics question, but Desmos is very helpful in solving and testing mathematics issues, so maybe anyone could help me. I can't figure it out how to use AND condition ...
15
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2answers
371 views

Power series which diverges precisely at the roots of unity, converges elsewhere

Is there a complex power series $\sum a_nz^n$ with radius of convergence $1$ which diverges at the roots of unity (e.g., $z=e^{2\pi i\theta}$, $\theta \in \mathbb{Q}$) and converges elsewhere on the ...
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5answers
130 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$.

I am analizing the convergence, absolute convergence and conditional convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$. I proved already that the series $\sum_{n=1}^{\...
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0answers
51 views

Multiplying conditionally convergent series. Is this valid?

Normally when I see this form of the product of infinite series: $$ \left(\sum_{n=0}^\infty a_n\right)\left(\sum_{m=0}^\infty b_m\right) = \sum_{n=0}^\infty\sum_{m=0}^\infty a_nb_m $$ I see the ...
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About relatively uniform convergence in Riesz spaces

A Riesz space is a partially ordered real vector space such that every two elements has a sup and indeed it is a vector lattice. I have encountered to the definition of relatively uniform convergence ...
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0answers
23 views

A Question on Convergence in Probability of Conditional Characteristic Functions

This post is a bit long, so please don't be annoyed with me :). I have a question regarding convergence in probability of conditional characteristic functions, which I describe below: Let $X_n$ be a ...
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2answers
52 views

If series of terms $a_j, b_j$ are convergent, must $\sum_j^\infty a_j b_j$ be convergent?

If $\sum_j^\infty a_j$ and $\sum_j^\infty b_j$ are convergent, show that $\sum_j^\infty a_j \cdot b_j$ is convergent or provide a counter example where it is not convergent. If it can be assumed ...
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0answers
34 views

A Question about the Convergence of Conditional Distributions

I have a question regarding weak convergence of conditional distributions in a specific setting, that I require for my research. Suppose that $X_n$ and $Y_n$ are two sequences of random variables ...
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2answers
149 views

Rearrangement of conditional convergent series [closed]

Given - 'For given conditional convergent series, it can converges to any number R with suitable rearrangement' So how can we say that series is convergent, as it can be made to converge to any ...
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1answer
24 views

Show that the given series is conditionally convergent.

Show that the series $$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n -\log n}$$ is conditionally convergent . My Work Let $a_n=\frac{(-1)^{n}}{n -\log n}$ then $|a_n|=\frac{1}{n -\log n}$ We can write ...
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1answer
75 views

find when $\sum^{\infty}_{n=0} x^n \tan \left(\frac {x}{2^n}\right)$ is convergent

For which real numbers x is the series $$\sum^{\infty}_{n=0} x^n \tan \left(\frac {x}{2^n}\right)$$ convergent and how (i.e. absolutely/conditionally)? I have proved that the series converges ...
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1answer
77 views

Convergence of complex integrals: Necessary and Sufficient conditions.

Currently I am examining functions defined in the following way: $F(z)=\int f(z,t) dt\ $where the integral is along some curve $\gamma\\$ not necessarily closed. I want to know necessary and ...
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1answer
34 views

Determine whether the given series is absolutely convergent or conditionally convergent

Consider the series $$\sum_{n=1}^\infty \log\left(1+\frac{1}{|\sin(n)|}\right).$$ Determine whether it converges absolutely or conditionally. I am trying to apply Cauchy condensation test, but I ...
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311 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
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1answer
37 views

Absolute and conditional convergence of series with parameter

I have the following series: $$\sum_{n=2}^{\infty}\frac{(-1)^n}{(n+(-1)^n)^p}$$ I need to check for absolute and conditional convergence values, depending on the parameter $p$. Any tips on how to ...
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4answers
78 views

Sum of conditionally convergent series

The series $\sum_{n=1}^{\infty}2^{-n}$ is absolutely convergent. It converges to $\frac{1}{1-\frac{1}{2}}$. Series $\sum_{n=1}^{\infty}(-1)^{n}(2^{-n}+n^{-1})$ is convergent, but only conditional ...
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1answer
43 views

A counterexample to monotonicity in Leibniz criterion for alternatig series

Leibniz criterion for alternating series says that the series $$ \sum_{n=0}^{+\infty}(-1)^n a_n $$ conditionally converges if the following two hypothesis are verified: $\lim_{n\to+\infty}a_n =0$, $...
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1answer
85 views

Conditional convergence of a series involving $sin n \theta$

I recently stumbled upon the series $$\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin n \theta}{n}.$$ Consider all values of $\theta$ except $k \pi$ where $k$ is an integer. ...
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2answers
52 views

Is there an intuitive way of thinking why a rearranged conditionally convergent series yields different results?

We know that any conditionally convergent series can be made to converge to anything, or even diverge. My question is if we can intuitevely explain why such a thing happens. One would be tempted to ...
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2answers
290 views

When $\Big[ uv \Big]_{x\,:=\,0}^{x\,:=\,1}$ and $\int_{x\,:=\,0}^{x\,:=\,1} v\,du$ are infinite but $\int_{x\,:=\,0}^{x\,:=\,1}u\,dv$ is finite

I have encountered a simple problem in probability where I would not have expected to find conditional convergence lurking about, but there it is. So I wonder: Is any insight about probability to be ...
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0answers
36 views

Modified alternating harmonic series [duplicate]

I am having trouble characterizing the types of alternating with-differing-period harmonic series which converge. For example, $\sum_{n\in \mathbb{N}} (-1)^n\frac{1}{n}$ converges, but what about $1+\...
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1answer
63 views

How do we rearrange the terms of the harmonic series so they add up to 0

The harmonic series converges conditionally; therefore, the series can be rearranged in any way to get different sums, but how do we rearrange in such a way that it equates to 0. Is there a trick/ ...
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1answer
102 views

Conditional Convergent Power Series and radius of convergence

I'm just not sure about the fifth case which states that CC at L and CC at R, I've failed to find an example for that, I cannot prove it is wrong either, can someone give me some hints?
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0answers
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Will the average of a conditional sample converge in probability to the conditional expectation?

Suppose I have a random variable $X$ from which I sample $n$ times. Then I take the average of only those realizations that fullfill a certain condition $A$. Will this average converge in ...
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4answers
384 views

Why does the commutative property of addition not hold for conditionally convergent series?

I learned about the Riemann rearrangement theorem recently and I'm trying to develop an intuition as to why commutativity breaks down for conditionally convergent series. I understand the technique ...
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1answer
103 views

Conditional convergent improper Riemann integral vs. Lebesgue Integral

For all this, I'm thinking on functions defined on $\mathbb{R}$. I've already read that if a function $f$ is absolute improper Riemann integrable, then $f$ is Lebesgue integrable and both integral ...
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2answers
324 views

Every non absolutely convergent series can be rearranged to converge to any $\lim \sup / \inf$ (Rudin)

Questions: 1- Why does the divergence of either one $(\sum p_n, \sum q_n)$ and the convergence of the other imply the divergence of $\sum a_n$ while the divergence of both of them doesn't imply the ...
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0answers
153 views

Big O analysis and order of convergence summarized

I want to confirm my understanding about rate of convergence and Big O analysis, where the argument inside of $O$ is a function of the error between $X_n$ and what it's converging to $L$. My ...
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0answers
26 views

System stability and $z$-transform

I have been struggling to understand the relation between $z$-transform and the study of analysis but there is something that puzzles me. Stability of a discrete time system is happens if and only if ...
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2answers
56 views

Check if the series $\sum_{n=1}^\infty\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}$ converges

Check if the series $\sum_{n=1}^\infty\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}$ converges So I think that the series is conditionally convergence that what I did so far $$\sum_{n=1}^\infty\vert\frac{(-...
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3answers
128 views

Need a second help in understanding a step in matrix representation of bounded linear operators.

In completion to this question: Need A help in understanding a step in matrix representation of bounded linear operators. The book said: "Now, $$A \phi_{j} = \sum_{k}<A \phi_{j},\phi_{k}> \...
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2answers
41 views

Infinite series convergence question

$$\sum_{n=3}^{\infty}\frac{(-1)^n}{\log n}$$ Can the conditional convergence of this series be proved by alternating series test, since you need n to be a natural number for the alternating series ...
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1answer
207 views

Theorem 3.54 Baby Rudin (Riemann's Series Theorem) [duplicate]

I am struggling trying to understand the final part of the proof of Theorem 3.54 on Baby Rudin. Here's the Theorem Let $\sum a_n$ be a conditionally convergent series. Suppose : $$ -\infty \leq \...
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1answer
94 views

Convergence of alternating series (not monotone)

I'm dealing with the following series $$\sum_{n\ge 2} \frac{1}{2+(-1)^nn}\,.$$ Clearly this will not converge absolutely as the general term in absolute value $\sim 1/n$. Leibniz can not be applied ...