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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34 views

Computation of the convergence radius of special power series

As we know, for the computation of the convergence radius of the power series, we have two method, one is ${R=\lim _{n \rightarrow \infty }\left|{\frac {a_{n}}{a_{n+1}}}\right|}$,the other is $R = \...
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1answer
48 views

Power series that converges conditionally at the endpoints of its interval of convergence

As we know, for the power series $$\sum_{n=1}^{\infty}a_{n}(x-x_0)^n,$$ it will have a convergence radius $R$ and $R$ is non-negative. I want to know whether there exists a power series example that ...
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1answer
64 views

Can (the partial sums of) a conditionally convergent series always be written as an alternating sequence of decreasing terms?

True or false: If $\ \sum a_n\ $ is conditionally convergent series, then there exists an increasing sequence of integers $\ k_1,\ k_2,\ k_3,\ldots\ $ such that $$\ \left(\ \displaystyle\sum_{n=1}^{...
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1answer
42 views

Can a power series conditionally converge outside its radius of convergence?

Having learned about conditional and absolute convergence of series, I was confused when the methods used to find the interval of convergence of a power series seemed to consider only absolute ...
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1answer
32 views

$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure

For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
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25 views

Condtitional and absolute convergence of integral

Check for absolute and conditional convergence $\int_0^2 \frac{\sin\frac{1}{\sqrt x}}{x^\alpha\ln x}$.
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36 views

Determining if the function is conditionally convergent or absolutely convergent

The given series is $$\sum^{\infty}_{n=1} \frac{\sin(\frac{\pi n}{2})}{{n}^{\frac{2}{3}}}.$$ I have observed that the given is equivalent to $\sum^{\infty}_{n=1} \frac{\sin(\frac{\pi n}{2})}{{n}^{\...
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31 views

Convergence with asymptote

Series is given as $$\sum_{m=2}^{\infty}\frac{\cos(\pi m)}{\ln(\ln(m))}$$ How does it converge? I have shown that it diverges for the absolut series by comparison to the harmonic series. i.e $$\frac{...
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1answer
50 views

Convergence of a alternating series (tip of the tongue)

Let a series be given as $$\sum_{m=2}^{\infty}\frac{\cos(\pi m)}{\ln(\ln(m))}$$ Is it converges conditionally, converges absolutely or diverges. Attempt I have found out that the series does not ...
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38 views

A question related to the Alternating Harmonic Series [duplicate]

It's a well known result that the Alternating Harmonic Series : $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... = \ln(2)$$ But, this is because of "Conditional Convergence". I saw ...
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72 views

Determine convergence of a series

Let a series be given as $$\sum_{m=2}^{\infty}\frac{\cos(\pi m)}{\ln(\ln(m))}$$ Is it converges conditionally, converges absolutely or diverges. Attempt From computation I see that it does indeed ...
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17 views

Sub sequence of Cauchy sequence and its sum

i want to take a good sub-sequence out of a Cauchy sequence. Let $(x_n)$ be cauchy sequence , then there exist sub sequence $(x_{n_k})$ where $\sum d(x_{n_k},x_{n_{k+1}}) < \infty$ is satisfied. My ...
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20 views

Interval of Convergence of Laplace Transform.

Consider the set of conditions: $X(s)$ has exactly two poles. $X(s)$ has no zeros in the finite $s-$plane. $X(s)$ has a pole at $s=-1+j$. $e^{2t}x(t)$ is not absolutely integrable. $X(0)=8$ Alright ...
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Why do we ask for *absolute* convergence of a series to define the mean of a discrete random variable?

If $X$ is a discrete random variable that can take the values $x_1, x_2, \dots $ and with probability mass function $f_X$, then we define its mean by the number $$\sum x_i f_X(x_i) $$ (1) when the ...
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31 views

Cauchy product of an absolutely convergent sereis and a conditionally convergent series [duplicate]

We know by Mertens' theorem, Let $(a_n)_{n≥0}$ and $(b_n)_{n≥0}$ be real or complex sequences, then if $\sum_{n=0}^{+\infty}a_n$ converges absolutely and $\sum_{n=0}^{+\infty}b_n$ converges only ...
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14 views

Conditional convergence of an infinite series of logarithms

Let $\rho_n$ be a sequence of positive real numbers converging to $0$. Suppose that the following series is convergent $$ \sum\log(1 + \rho_n(2\cos(\theta_n) + \rho_n)). $$ We may assume that the ...
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1answer
40 views

Combining terms in a conditionally convergent series

I am aware that one is unable to rearrange terms in a conditionally convergent series. But, take a conditionally convergent series, say $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$ and group terms ...
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1answer
63 views

If $\int_{a}^{\infty} f(x)dx$ and $\int_{a}^{\infty} g(x)dx$ converge conditionally, does $\int_{a}^{\infty} f(x)g(x)dx$ absolutely converge?

My question is: Are there any two functions $f(x)$ and $g(x)$ Which hold that $\int_{a}^{\infty} f(x)dx$ and$\int_{a}^{\infty} g(x)dx$ converge conditionally but $\int_{a}^{\infty} f(x)g(x)dx$ ...
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1answer
47 views

Alternating series - determine if it converges absolutely, conditionally or diverges using alternating p-series test

I am trying to determine to whether the series $\sum _{n=0}^{\infty }\:\left(-1\right)^n\frac{1}{\sqrt{n}\left(ln\left(n\right)^{2021}\right)}$ is conditionally convergent, absolutely convergent, or ...
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34 views

Does $E[\hat{F}_n(X_1) - F(X_1) \mid \cdots]\to 0$?

Let $\binom{X_1}{Y_1}, \binom{X_2}{Y_2} \dots $ be iid vectors. Denote $\hat{F}^x_n(\cdot) = \frac{1}{n}\sum_{i=1}^n1[X_i\leq \cdot]$ an empirical distribution function of $X$, and similarly $\hat{F}^...
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Is it possible for a power series to be conditionally convergent at two different points? [closed]

Like I stated in the title, I was just wondering if it's possible for a power series to be conditionally convergent at two different points. Are there any examples of power series that fit this ...
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57 views

Examine for conditional and absolute convergence: $\int_1^\infty \frac{\cos y\,\mathrm{d}y}{(y+\sin y)^\alpha}$

I need to examine the integral $$\int\limits_1^\infty \frac{\cos y \,\mathrm{d}y}{(y+\sin y)^\alpha}$$ for conditional and absolute convergence based on $\alpha$'s values. Since this integral looks ...
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1answer
114 views

Proof that $\lim_{n \to \infty}\sqrt[n]{|a_n|}=1$.

Suppose that there is an arbitrary $a_n$ and that $\sum_{n=1}^{\infty}a_n$ is a conditionally convergent series. Then prove if $\lim_{n \to \infty}\sqrt[n]{|a_n|}$ exists, then $\lim_{n \to \infty}\...
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24 views

summing basis of c_0 is a conditional basis

The standard unit vector basis $\{e_n\}$ is an unconditional basis of $c_0$ and $l^p$ for $1 \leq p < \infty.$ An example of a Schauder basis that is normalized conditional (i.e., not ...
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2answers
50 views

Prove if $a_n$ is non-negative and $\sum a_n$ converges then $a_n\leq\frac{1}{n}$ for all $n\geq N$ for some some N

I believe this is intuitively true but I cannot figure out how to prove it. I was thinking I could use a proof by contradiction. If there is no such $N$ then there are an infinite number of terms of $...
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24 views

Interesting natural rearrangements of conditionally convergent sums?

In my first course in analysis, we briefly covered the proof of the Riemann Rearrangement theorem as an exercise, from which we are assured that we could rearrange the alternating harmonic series to ...
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18 views

Conditional convergence and Riemann rearrangement theorem

I have read that a series is conditionally convergent if it's absolute sum diverges while it converges. However I've also read that after suitable rearrangement, conditional convergent limit can be ...
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1answer
27 views

Convergence of a series where terms involve a convergent sequence

I have been given the following question: Let $ (x_n)_{n \in \mathbb{N}} $ be a sequence of real, positive numbers such that $ x_n \rightarrow L $ as $ n \rightarrow \infty $. If $L>0$, prove that ...
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2answers
181 views

For any conditionally convergent series $\sum _{n=1}^\infty a_n,\ \exists\ k\geq 2\ $ such that the subseries $\sum _{n=1}^\infty a_{nk}$ converges.

A subseries of the series $\displaystyle\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\displaystyle\sum _{k=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove or disprove: For any ...
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2answers
45 views

Conditional expectation of a bounded almost sure random variable

Let $Z$ a random variable which is $\mathscr{p}$-measurable and bounded almost sure, ie, there is a positive number $M$ such that $|Z| \leq M $ a.s.Then show thatfor the conditional expectation $$Y = ...
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1answer
54 views

conditional expectation of disjoint events

Let $B_{1},B_{2},\dots,B_{n} \in \mathcal{F}$ partition of $\Omega$ and $\mathcal{P}=\sigma(B_{1},B_{2},\dots,B_{n})$ If $P(B_j)>0 , j = 1,\dots,n$. Then show that : $$\mathbb{E}[X|\mathcal{P}] = \...
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1answer
66 views

Follow-up question on conditionally convergent series.

This is a follow-up from this question. Hypothesis: If $\sum\limits_{n=0}^{\infty} a_n\ $ is a conditionally convergent series with $\ \limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = 1$, ...
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3answers
69 views

Is there an ambiguity on $\sum\limits^{\infty}_{i=0}a_i$?

Sorry if this is basic but I saw the theorem that says if $\sum\limits^{\infty}_{i=0}a_i$ is conditionally convergent then this series can be any number up to rearrangement. Then does it imply that $\...
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1answer
52 views

Prove specially rearranged alternating harmonic series converges to $\frac 12 \ln{\frac{4p}{q}}$

By Leibnitz's test the alternating series is convergent. $\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac 1n =\frac 12 \ln{\frac{4×1}{1}}$ \begin{align} & \left(1-\frac 12- \frac 14\right)+\left(\...
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1answer
61 views

How can I show that if 2 power series are conditionally convergent then it can happen that $\sum_{k=1}^{\infty}c_k$ diverges?

Assume that $\sum_{k=1}^{\infty}a_k =A \in R$ and $\sum_{k=1}^{\infty}b_k =B \in R$ with partial sums $r_n =\sum_{k=1}^{n}a_k$ and $s_n=\sum_{k=1}^{n}b_k.$ If one wants to sum the double series $\sum_{...
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1answer
46 views

Absolute convergence of $\sum_{n=1}^{\infty} a_nx^n$

I have to find an $a_n$ such that $\sum_{n=1}^{\infty}a_nx^n $ converges absolutely on $[-1,1]$. I have choosen $a_n = \frac {1}{n^2}$ and did the root test and see that $\lim_{n \to \infty} (|a_n|) =...
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1answer
63 views

if $\sum_{i=1}^n a_i$ converges but $\sum_{i=1}^n (a_i)^2$ diverges, does that mean that $\sum_{i=1}^n a_i$ is conditionally [duplicate]

I don't think this is true, but cannot find an example to disproof it. Either that or i need to prove that $\sum_{i=1}^n |a_i|$ diverges, which I am unclear how to approach the proof.
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141 views

Prove that the series $\sum\limits_{n=2}^{\infty}(-1)^n\frac{\ln(n)}{n^x}$ converges to a positive real number for all $x > 0$

Prove that the series $\sum\limits_{n=2}^{\infty} (-1)^n\frac{\ln(n)}{n^x}=\frac{\ln(2)}{2^x} - \frac{\ln(3)}{3^x} + \frac{\ln(4)}{4^x} - \frac{\ln(5)}{5^x} + ...$ converges to a positive real number ...
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47 views

Decidability of convergence of real series given an oracle for positive series

I've been reading for the last hour a few posts on this site about series that no one knows if they converge or not. I was quite surprised that they were almost all consisting of positive terms. ...
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1answer
195 views

Is the series conditionally convergent, absolutely convergent or divergent $\sum(-1)^n\frac{\ln^3 n}n$.

Determine if the series $$\sum_{n=1}^\infty(-1)^n\frac{\ln^3 n}n$$ is conditionally convergent, absolutely convergent or divergent. By comparison test I got $a_n > b_n$ therefore divergent with $...
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3answers
38 views

Absolute vs Conditional Convergence of an alternating series [closed]

Picture of the Alternating Series $$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt[4]n(\sqrt{n+2})}$$ This is the picture of the alternating series I am working with. I found that it is convergent by the ...
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1answer
132 views

The alternating series $\sum_{k=1}^{\infty} \frac{(-1)^k(2 - \sin k)}{2k}$ seems to be convergent, but Leibniz criterion does not apply

I was looking for an example of convergent, alternating series $\sum_{k=1}^{\infty}(-1)^kb_k$ such that $\{b_k\}_{k=1}^{\infty}$ is not eventually monotone, so that Leibiniz criterion could not be ...
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0answers
98 views

Laplace Transform of $\cos(t)/t$

this seems like a homework problem. yes! To some extent. But really I was not getting it. I was not able to get the Laplace transform of $\cos(t)/t$. using the property of Integration in Laplace ...
4
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1answer
314 views

Convergence of $\sum\frac{\sin n\theta}{n^r}$ and $\sum_{n=1}^\infty u_n \cos (n\theta+a)$.

Problem 1. Show $q$th power of $\sum\frac{\sin n\theta}{n^r}$ (formed by Abel's rule, i.e. $$\nu_n=\sum_{i_1, i_2,\dots,i_q=n} \frac{\sin i_1\theta}{{i_1}^r}\dots\frac{\sin i_q\theta}{{i_q}^r},$$ ...
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1answer
36 views

Proving convergence in probability functions

In the textbook during the steps to expand the convergance of probability the following is provided. the condition for converges in probability is given as $\lim_{n\rightarrow \infty}\mathbb{P}(|X_n-X|...
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2answers
151 views

Conditional convergence for improper Riemann double integrals

I'm reading Buck's advanced calculus. It says for improper integral of higher dimensions, conditional convergence is impossible, i.e., $\int\int_D f$ cannot exist without $\int\int_D|f|$ existing too. ...
5
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207 views

Convergence of Euler product implies convergence of Dirichlet series?

(Crossposted to Math Overflow) Suppose we have an Euler product over the primes $$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$ where each $a_p \in \mathbb{C}$. The Euler product is ...
4
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1answer
152 views

If $\sum a_n^k$ converges for all $k \geq 1$, does $\prod (1 + a_n)$ converge?

By definition, an infinite product $\prod (1 + a_n)$ converges iff the sum $\sum \log(1 + a_n)$ converges, enabling us to use various convergence tests for infinite sums, and the Taylor expansion $$ \...
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1answer
52 views

Convergence of a rearrangement of conditionally convergent series

$\{a_n\}$ is a sequence of real numbers.$\space\sum_{n=1}^{\infty} a_{2n}$ and $\sum_{n=1}^{\infty} a_{2n-1}$ are both conditionally convergent. Is there such $\sum_{n=1}^{\infty} a_{n}$ that is ...
2
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0answers
46 views

What is the domain of convergence for the Taylor Series for this function?

What is the domain of convergence in variable $a$ of the Taylor Series of this function? $$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)ln\left(p\left(x,a\right)\right)dx$$ ...