Questions tagged [absolute-convergence]
This tag is for questions related to absolute convergence of a series.
698
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If a series converges absolutely over the real numbers, then does it absolutely converge over the complex numbers too?
It was stated at the beginning of my lecture notes that the Taylor expansion of the sine, cosine, and exponential function converge, i. e. we have
$$\sin{x} = \sum_{\nu = 0}^\infty (-1)^\nu \frac{x^{2\...
- 109
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Test the convergence of the series $\sum_{n=2}^{\infty} \dfrac{(-1)^{n+1}}{n\space log\space n}$ [duplicate]
So I encountered this problem in S. K. Mapa's book Introduction to Real Analysis. The answer to this question is given as below:
The series is $\sum_{n=2}^{\infty} \dfrac{(-1)^{n+1}}{n\space log\...
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1
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Travel from A to B in finite number of days?
Suppose, the distance between city A and city B is 1. Each day, I complete $90\%$ of the residual distance. For example, on day 1, I will travel 0.9, on day 2, I will travel $90\%$ of the left ...
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Clarification on a remark by Schaefer on nuclear spaces and absolute versus unconditional summability
In my (poor) attempt to answer Unconditional and absolute convergence in non-Banach spaces, essentially asking about whether or not we can get rid of the completeness assumption in the Dvoretzky-...
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Unconditional and absolute convergence in non-Banach spaces
I know that, by Dvoretzky-Rogers theorem, we know that in a Banach space $X$ the following are equivalent:
$X$ is of finite dimension.
Every unconditonally convergent series is absolutely convergent.
...
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3
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Solving $\int_{0}^{1}\sin\lfloor\frac{1}{x}\rfloor dx$
This is just for fun
I know that without the floor function, the solution to this integral would be $\sin{1}-\operatorname{Ci}{1}$
My first idea to solve this is by creating an infinite summation.
$$\...
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2
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Why isnt the sum of the results of random trials based on a random variable the same as the expected value of the random variable time the # of trials
If $X$ is a random variable and $X_i$ the ith result of an experiment whose underlying probability distribution is $X$ then by the law of large numbers
$$\lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{...
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On Convergence of Alternating Harmonic Series
I am almost new with Mathematical Analysis and I see something that made me to think! It is proved that Alternating Harmonic Series is convergent to ln(2). What if ...
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Proof that a series converges to zero
I am working on the following problem arising in time series analysis.
Let us assume that $\sum_{h \in \mathbb{Z}} |\gamma(h)|<\infty$. I would like to prove that
\begin{equation*}
1) \; \; \; \...
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Sufficient condition to apply integration by parts "infinitely many times"
My question is related to this one.
Suppose we are trying to solve the following integral
$$ \int f(x) g(x) dx,$$
where we know $f(x)$ is smooth, all of its derivatives are positive, and the sum of ...
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59
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Necessary and Sufficient Condition for A Particular Sum Rearrangement
Let $\phi:\mathbb{N}\rightarrow\mathbb{N}$ be a rearrangement of $\mathbb{N}$ (a bijection). I am searching for a condition equivalent to:
$$$$
For all complex sequences $(\alpha_n)$, there exists ...
- 399
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What happens if I change the norm of an absolute convergent series?
Let us consider the topological vector space $\mathbb{C}$ equipped with the euclidean topology.
We know that a series of complex numbers $\sum_{n=1}^{\infty}a_n$ is said to be absolutely convergent ...
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Question about proof of the truncated Perron's formula dealing with bounds and convergence
I have a question about the proof of the truncated Perron formula in my analytic number theory lecture notes.
The Formula is given as follows: Let $x,c,T>0$ and suppose that $\sum_n |a_n|/n^c$ is ...
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Vector-Autoregression: Assertion on the convergence radius of a power series with square-matrices as coefficients
I first want to give some context to understand the setup of my question (but you may provide an answer without knowing anything about time series analysis - I guess).
Anyways: In a proof that derives ...
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How would you prove $\sum _{n=2}^{\infty } (-1)^n \left(n^{1/n}+\eta '(n)-\frac{\log (n)}{n-1}-1\right)+\eta '(1) =$ the MRB constant
The step I'm having trouble with is
$\sum _{n=2}^{\infty } (-1)^n \left(n^{1/n}+\eta '(n)-\frac{\log (n)}{n-1}-1\right)+\eta '(1)
=-\left(\sum _{m=2}^{\infty } \frac{(-1)^m \eta ^m(m)}{m!}+\eta '(1)\...
- 291
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2
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How is absolute convergence used in "if a series converges absolutely, then every rearrangement converges to the same limit?"
Here is Abbott's proof:
Assume $\sum\limits_{k = 1}^{\infty} a_k$ converges absolutely to $A$, and let $\sum\limits_{k = 1}^{\infty} b_k$ be a rearrangement of $\sum\limits_{k = 1}^{\infty} a_k$. Let'...
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Does absolute convergence of $\sum_{n=1}^{\infty} a_n$ implies $(a_n)^{1/n}$ tends to some $r \in [0.1]$?
True or false: Let $\sum_{n=1}^{\infty} a_n$ be an absolutely convergent series, then $(a_n)^{1/n} \rightarrow r \in [0.1]$?
My initial progress: evidently $|a_n|\to 0$,so eventually $|a_n| ＜1$ and ...
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Convergence in prob. + Uniform integrability => L1 convergence (proof with rate of convergence) [closed]
We know that:
I am wondering if it is possible to be more precise and state that assuming the conditions of the theorem and adding that
$P\left(\left|X_n-X\right| \geq \epsilon\right) \leq C/n$.
...
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Absolute and unconditional convergence in $C[0,1]$
I'm trying to come up with example of series from $C[0,1]$ that converges unconditionally, but not absolutely. I know for sure that it exists, but I can't find an example. Firstly I came up with idea ...
2
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Investigate the absolute convergence of the integral
I want to investigate the absolute convergence of integral.
$$\int_{0}^{\infty} \; x^4 \; \sin(e^{2x}) \; dx$$
I made a replacement
$$t = e^{2x},\; x = \frac{\ln{t}}{2} \\
\int_{1}^{\infty} \; \frac{...
- 27
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1
answer
80
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Show a sum in a Banach space is convergent and bounded
$f,g \in l^2(\mathbb{Z})$ such that $\sum_{n \in \mathbb{Z}} |f(n)|^2 < \infty, \sum_{n \in \mathbb{Z}} |g(n)|^2 < \infty$.
Show that $(f \star g)(n) = \sum_{m \in \mathbb{Z}} f(m)g(-m+n)$ is ...
user781460
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0
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convergences of series in two different metrics
for the past few days I've been studying the topic of metric spaces and now making some exercices on it. However I'm struggling with the following exercice: let $l^1(\mathbb{N})$ denote all function $...
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The limit is independent of the order of summation if the series converges absolutely?
I'm trying to find an explanation and an example that proves that if the series converges absolutely, then the limit is independent of the order of summation.
If anyone can help with this one I would ...
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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?
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Conditional/Absolute Convergance
The series $\displaystyle \sum_{n=0}^\infty a_n(x-2)^n$ converges conditionally at $x=-1$ and diverges at $x=7$. Which of the following could be true?
A) The series converges absolutely at $x=5$.
B) ...
2
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1
answer
71
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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 ...
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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 ...
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Test absolute convergence of series $\sum_{k=1}^{\infty} (\frac{1}{k} - \frac{1}{k!})$
So I need to test if the series $\sum_{k=1}^{\infty} (\frac{1}{k} - \frac{1}{k!})$ is absolutely convergent or not. So far, I've decided to go ahead with the ratio test, actually with a corollary of ...
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Test the absolute convergence of the series $\sum_{k=1}^{\infty}(-1)^k\frac{\ln(k)}{k}$
I need to test the following sequence for absolute convergence:
$$\sum_{k=1}^{\infty}(-1)^k\frac{\ln(k)}{k}$$
but I think I'm missing something. My comparison approach would be:
Disregard $(-1)^k$ ...
0
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1
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60
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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 ...
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Convergence of generalized hypergeometric function for the case $p=q+1$
it is known that the generalized hypergeometric function is defined by:
\begin{equation}
\label{e:pFq}
{}_{p}F_q(\textbf{a};\textbf{b};z)=\sum_{n=0}^{\infty}\frac{(a_1)_n\cdots (a_p)_n}{(b_1)_n\...
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0
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35
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Ratio test for multivariate power series
I wonder if there is a generalization of the ratio test to series in several variables. However, I have not been able to find any comprehensive literature on this question. Starting from a ...
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2
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93
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Proving Absolute Convergence of $\sum_{n=1}^{\infty} (-1)^n\ln \left [ 1+\sin \left (\frac{\pi}{n\sqrt{n}} \right) \right ]$
I am trying with no success to prove the Absolute/Conditional Convergence / Divergence of the following series:
$$\sum_{n=1}^{\infty} (-1)^n\ln \left [ 1+\sin \left (\frac{\pi}{n\sqrt{n}} \right) \...
- 67
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1
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Hardy class of bounded analytic functions is Banach space
I need help with the following:
Let $H^{\infty}(\mathbb{D})$ denote Hardy class of bounded analytic functions on unit disc $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$. Prove that $$||f|| = \...
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Generalization of the lemma, which connects absolute convergence and unconditional convergence of the series in Hilbert Space
I consider equivalence of absolute convergence and unconditional convergence in Hilbert space for $X = \mathbb{C} $
Lemma: If $(c_{n})$ is a sequence of real or complex scalars, then
$\sum_n c_n$ ...
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1
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Understanding the relation between transitive closure and convergent series
I am trying to better understand the concept of a transitive closure of a relation in the infinite case.
Suppose we have the set {$ q \in \mathbb{Q}| \exists n \in \mathbb{N}, q = 1 - (\frac{1}{2})^n ...
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From pointwise summability to absolute summbability in $L^\infty$, approximately
Let $(X,\Sigma,\mu)$ be a probability space and $(f_n)_{n \in \mathbb{N}}$ a sequence of measurable functions $f_n : X \to [0,1]$. Suppose that $\sum_n f_n$ converges pointwise, and let $\varepsilon &...
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Absolute summability implies square summability
I am aware of the fact that $\ell^1(\mathbb{R})$ is a subset of $\ell^2(\mathbb{R})$, but I somehow can't find a very good proof of absolute summability implying square summability across the internet....
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Does convergence of series imply absolute convergence on some disk
In Serge Lang's complex analysis book there are some theorems that relates convergence with absolute convergence (in the harder direction) ,e.g
If $f(z) = \sum a_nz^n $ has radius of convergence $r&...
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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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A question about convergence of functions involving Q-Pochhammer symbols
Consider the function $H:\mathbb{C}\rightarrow\mathbb{C}$:
$$H(s)=\prod_{n=1}^{\infty}\left(1+\frac{1}{2^{ns}}\right)$$
$H(s)$ is convergent on $\Re(s)>0$. Is it absolutely convergent on $\Re(s)>...
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Defining absolute convergence for sums over countable sets
I recently came across the following definition while reading on wikipedia:
Suppose $X$ is a countable set and $f:X\to\mathbb{R}$ is a real-valued function. Then we have that $\sum_{x\in X}f(x)$ is ...
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Change of order of integration with the inner integral being improper and uniform convergent on the integration segment of the outer integral
Suppose $f(x),\phi(x)$ are absolutely integrable on $(-\infty, \infty)$ and $g(\lambda)$ is the Fourier transform of $f(x)$. So we have that $f(x) = \dfrac{1}{2\pi}\int_{-\infty}^{\infty}{g(\lambda)e^{...
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In calculus, absolute convergence implies convergence. Why is the proof so weirdly complicated?
Here is the standard proof I got from Paul's math notes
First notice that $\left| {{a_n}} \right|$ is either ${a_n}$ or it is
$- {a_n}$ depending on its sign. This means that we can then say,
$$0 \le ...
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0
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Let $x_n, a_n$ sequences. Prove $\sum_{n=1} ^{\infty} a_nx_n$ converges when $\sum_{n=1} ^{\infty} ||x_n||$ conv. & $\limsup_{n\to\infty}|a_n|<\infty$
Real Analysis.
Hello, im taking a Real Analysis course and I'm trying to finish this proof given as an exercise.
Question:
Let $(X,||\cdot||)$ be a normed space. Let $x_n \subseteq X, a_n \subseteq \...
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Convergence of $\sum_{\omega \in \Lambda\setminus \{0\}} \frac{1}{\omega^k}$ for $k > 2$ and any lattice $\Lambda\subset \Bbb C$
This post is about the absolute convergence of the Eisenstein series (see these notes), i.e., for any lattice $\Lambda \subset \mathbb C$ and $k > 2$,
$$\sum_{\substack{\omega\in \Lambda\\\omega \...
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Can I use comparison test in this situation?
When you are given only that $\sum a_k$ converges but you're not sure if it converges absolutely, and you need to prove some related series $\sum b_k$ also converges, should I probably not use ...
- 349
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1
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54
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Why does $E|X|=\infty$ imply $EX=\infty$?
In Casella and Berger Example 2.2.4(Cauchy mean), in trying to show that Cauchy random variable $X$ has no mean, the authors prove that $E|X|=\infty$. Since we want to prove that $EX=\infty$, it has ...
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Absolute convergence of series with cosines
Consider the series:
\begin{equation}
\sum\limits _{n=1}^{+\infty}\cos\left(nx\right)\cos\left(ny\right),\quad x,y \in \left(0,\pi\right)
\end{equation}
What is the easiest way to prove that the ...
2
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1
answer
175
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Why does absolute convergence allow the order of summation to be changed? (Laurent expansion of the Weierstrass $\wp$-function)
I'm working through the derivation of the Laurent expansion of the Weierstrass P function (Theorem 1.11) in Tom Apostol's Modular functions and Dirichlet series in Number Theory.
The proof uses the ...
