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.
247 questions
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Limit of the function satisfying $f(x)=x-f(x^2)$ as $x\to 1^-$
I guess $\lim\limits_{x\to 1^-} f(x) = 1/2$, where the function $f(x)$ defined by $f(x)=x-f(x^2)$ in $[0,1)$, or by the series:
$$
f(x) = x - x^2 + x^4 - x^8 + x^{16} - x^{32} + \cdots.
$$
I know $f(x)...
- 81
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67
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A confusion with Cauchy theorem in the power series method of ordinary differential equations?
In my course of ODE(ordinary differential equation),the textbook introduce the power series method and give a uniquness theorem,Cauchy theorem.
Chapter 7: Power Series Solutions
The first person to ...
- 161
3
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1
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72
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Absolute/conditional convergence of $ \sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr) $
Examine the absolute/conditional convergence of the series
$$
\sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr)
$$
I can't seem to bound it from above, so I'm ...
- 437
1
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1
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64
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Validity of Continuous Mapping Theorem (Weak Convergence)
Suppose we have a pair of random variables $(X_n, Y_n)$, where $Y_n\geq0$. The pair $(X_n, Y_n)$ jointly converges to $(X ,Y)$, where $Y$ is a non-degenerate random variable and $P(Y>0)=1$.
Is it ...
- 408
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0
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29
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Convergence of an alternating series with mod 3-dependent decay [closed]
I would like to analyze the convergence of the following real series where an is defined as :
\begin{cases}
\displaystyle \frac{(-1)^n}{\sqrt{n}} & \text{if } n \equiv 1 \pmod{3}, \\\\
\...
8
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1
answer
160
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Given a sequence $y_n>0$ with $y_n\to 0.$ Does there exist a real sequence $x_n$ s.t. $\vert x_{n+1}-x_n\vert=y_n.$ and $\sum x_n$ converges?
Let $y_n$ be a sequence of non-negative real numbers such that $\displaystyle\lim_{n\to\infty} y_n = 0.$ Then does there exist a sequence $x_n$ of real numbers ("centred around $0$"), such ...
- 25.2k
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101
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Why is absolute convergence necessary for the Rearrangement Theorem?
The following proof of the rearrangement theorem is from Bartle and Sherbert's Introduction to Real Analysis, 3rd edition. I do not see where absolute convergence was used in the proof.
9.1.5 ...
- 25
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1
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38
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Optimal price converges on fixed proportion of consumer demand
I've been curious about revenue optimization and the proportion of a market that is served. If we suppose that customer $x$ is willing to pay $\le p$ dollars for product $y$, there is some optimal $\...
- 257
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103
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How do I find the abcissa of convergence of $f(n)=(-1)^{n-1}$?
I am trying to determine that the abcissa of convergence of $f(n)=(-1)^{n-1} $ is $\sigma_0(f)=0$ and the abcissa of absolute convergence is $\sigma_a(f)=1$ ,so we have the Dirichlet function : $f(n)...
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2
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245
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Formal derivation of $\int_0^{\pi/2}\frac{\log(1-\sin x)}{\sin x}dx=\frac{-3\pi^2}{8}$
EDIT: I realise that the question might have been not too clearly worded. I CAN do the computation with Feynmann's trick and have done so already, obtaining the correct result. What I CAN'T do (and ...
3
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2
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67
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Must the power sum of the sum of two infinite real sequences with convergent power sums also converge?
The motivation for this question was from a linear algebra problem asking to prove that the set of all real sequences with finite p-norm was a vector space, I did this using Jensen's on $x^p$, $p\ge 1$...
3
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1
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97
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Definite integral is a conditionally convergent sum
I'm considering this peculiar (improper?) integral: let $r(x) = \lfloor\frac{1}{x}\rfloor$, and $f(x) = (-1)^{r(x)} r(x)$. Consider:
$$\int_{0}^1 f(x) dx$$
I specifically machined the function $f(x)$ ...
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Is it possible to have a conditionally convergent Riemann sum?
The standard formula for a Riemann sum of an integral is as follows:
$$\int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i) \Delta x$$
where $\Delta x = \frac{b-a}{n}$ and $x_i = a+i\Delta x$.
...
- 682
4
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1
answer
152
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A series all of whose rearrangements are convergent, and yet not all rearrangements have the same sum?
Let $G$ be a normed abelian group1 (which seems to be the most general place to study rearrangements of series). For a sequence $x_0, x_1, \ldots\in G$, define
\begin{align*}
\Sigma & := \Bigl\{ \...
- 4,692
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1
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61
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Confused on double sum of non absolute convergent
I'm confuse and can't find my mistakes on the homework. The question asked to prove that
$$H(z):=\sum_{m\in\mathbb Z}\sum_{\substack{n\in\mathbb Z\\ (m,n)\not=(0,0),(1,0)}}\dfrac{1}{(m-1+nz)(m+nz)}=2-\...
- 1,118
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58
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Express $E[T \mid T\ge t, X]$ using conditional cdf $F(t\mid X)$.
Given $T$ is a nonnegative random variable and $X$ can be any arbitrary random variable. I want to rewrite $E[T \mid T\ge t, X]$ in terms of the conditional cumulative distribution of $T$, denoted as $...
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2
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71
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Find all the values $x\in\mathbb{R}$ where the Series converge
$$
\sum_{n=1}^{\infty}\frac{\sqrt{n}+3n}{2^{n}+5n}(x-1)^{n}
$$
I calculate the limit $n\to\infty$ with the D'Alambert ratio test, and the series converges in the interval $-1<x<3$:
$$
\sum_{n=1}^...
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71
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Proof verification: Show that if a series is conditionally convergent, then the series from its positive terms is divergent.
Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.
Please, help me to ...
- 1,777
1
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48
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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 ...
- 1,853
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1
answer
79
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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 ...
- 751
2
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1
answer
165
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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 ...
- 649
2
votes
0
answers
76
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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 $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
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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}^\...
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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 ...
- 11
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1
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72
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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 $$E[X \mid X \...
- 43
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52
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Convergence of Alternating Series Involving Cosine Term and Square Root
I am working on a series and attempting to determine its conditional convergence using the alternating series test.
$$
\sum_{n=1}^{\infty} \frac{\cos\left(\frac{\pi}{4} + 2\pi n\right)}{\sqrt{n}}
$$
I ...
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1
answer
57
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For conditionally convergent series $\sum a_n,\exists (m_n)_{n\in\mathbb{N}}$ with $(n-1)k<m_n\leq n k,$ s.t. $\sum_{n\in\mathbb{N}} a_{m_n}=\alpha.$
Properties of conditionally convergent series $\ \displaystyle\sum a_n\ $:
The sum of the positive terms is $+\infty;\ $ the sum of the negative terms is $-\infty.$
$\displaystyle\lim_{\substack{ { n\...
- 25.2k
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1
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Re-ordering conditionally convergent series over $\Bbb C$.
For a conditionally convergent series $\sum a_n$ over $\Bbb C$, let
$M\subset \Bbb N$ be a set such that the sub-series
$\displaystyle \sum_{n\in M} a_n$ converges absolutely.
$\sigma:\Bbb N \to \...
0
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1
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348
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What happens to EX if E|X| is infinity?
---------original question----------------
According to my professor, we can divide $X$ into $X_+=\max(X,0)$ and $X_-=-\min(X,0)$, both nonnegative.
And we have $EX=EX_+-EX_-$ and $E|X|=EX_++EX_-$
For ...
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2
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306
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Can a real function have convergence that oscillates depending on its derivative?
I recently read a post on here in which a user asked if there existed a function,
$$\lim_{x\rightarrow\infty}f(x)$$ is convergent, but;
$$\lim_{x\rightarrow\infty}f'(x)$$
does not converge.
I wanted ...
- 57
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1
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58
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How to prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent?
Let $f_n(x)=x^n+nx-1$, let $a_n$ denote its unique positive root. Then prove $\sum_{n=1}^{\infty}(-1)^na_n$ is conditionally convergent. Below is my solution.
First, $$a_n(a_n^{n-1}+n)=1,$$
because $...
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Is this proof that $\sum\cos(n)\frac{(n+1)^n}{n^{n+1}}$ converges conditionally, correct?
Determine if the following series converges absolutely, converges conditionally or diverges:
$$\sum\cos(n)\frac{(n+1)^n}{n^{n+1}}.$$
The series $\sum\cos n $ is bounded and the sequence $a_n=1/n$ is ...
- 2,262
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1
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36
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Convergence of a series given that the series with a specific parentheses insertion converges
I am given a sequence $(a_n)_n$ such that $|a_n|\le\frac{1}{n}$ and the sequence $b_k=\sum_{n=k^2}^{(k+1)^2-1}a_n$ and I am given that the series $\sum_k b_k$ converges. I need to show that the ...
- 71
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80
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On Convergence of Alternating Harmonic Series
I am almost new with Mathematical Analysis and I see something that made me to think! It is proved that Alternating Harmonic Series is convergent to ln(2). What if ...
- 111
2
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0
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67
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Necessary and Sufficient Condition for A Particular Sum Rearrangement
Let $\phi:\mathbb{N}\rightarrow\mathbb{N}$ be a rearrangement of $\mathbb{N}$ (a bijection). I am searching for a condition equivalent to:
$$$$
For all complex sequences $(\alpha_n)$, there exists ...
- 1,336
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0
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80
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how does rearranging terms of harmonic series result in different values?
In the book In Pursuit of Zeta-3 the author states that rearranging the terms of a harmonic series results in a different sum.
But isn't addition commutative? How ...
- 341
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54
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Conditionally convergent series value
Find all values of $p$ such that the series $\displaystyle\sum_{j=1}^{\infty}\dfrac{(-1)^j\cdot\log(j)}{j^p}$ is conditionally convergent.
I know that the alternating series test states that if a ...
- 11
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$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent arrangement prime number theorem related?
$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\log x}{x}}-1\right)$ Is its absolutely convergent neighbor prime number theorem related?
$ C_{MRB}=\sum _{x=1}^{\infty } (-1)^x\left(e^{\frac{\...
- 309
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0
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54
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Number of permutations for a conditional convergence series
I'm asking the next thing:
If we have a rational conditionally convergent series: $$\beta=\sum_{i=0}^\infty q_i \; \; \; ({q_i}\in\Bbb Q)$$
Then we know thanks to Riemann that it may be rearranged to ...
- 159
4
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1
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125
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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{...
- 115
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0
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32
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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?
2
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1
answer
156
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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 ...
- 23
0
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2
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111
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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 ...
- 10.4k
0
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1
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285
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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 ...
- 137
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0
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41
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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 ...
- 530
0
votes
1
answer
81
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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 ...
- 1,369
2
votes
1
answer
69
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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 ...
- 1,908
2
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1
answer
139
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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 ...
- 2,479
0
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1
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25
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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,
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0
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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}...
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