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Questions tagged [absolute-convergence]

This tag is for questions related to absolute convergence of a series.

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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}^...
lucasg638's user avatar
2 votes
0 answers
28 views

Existence of absolutely convergent subseries given the base sequence converges to zero

My Real Analysis final is coming up and I'd like to practice working with sequences and series, so I picked a practice problem and tried working it out. The statement is the following: Let $ (x_n)_n $...
simeondermaats's user avatar
-2 votes
1 answer
99 views

Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$

Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
Diego Martinez's user avatar
1 vote
1 answer
36 views

Bounding of this series [duplicate]

Prove this series converges absolutely. $$\sum_{k = 1}^{+\infty} \frac{(x^2-7x+6)^n}{n^2 6^{n+2}}$$ Attempts As in the comments, it doesn't say where. But I believe it's meant for $x \in (0, 3) \cup (...
Heidegger's user avatar
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23 views

How To Prove That The Absolute Convergence Of The One-Step Transition Matrix?

I am trying to prove that the n-step transition matrix in a Markov Chain satisfies the condition that the sum of each row is one (ie, each row is a valid probability distribution). I have done this ...
Azorbz's user avatar
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3 votes
1 answer
79 views

Proof of a Limit related to Gauss' Convergence test

So this is the question: if the series $\sum_{n=1}^{\infty} a_n$ is such that $$\frac{a_n}{a_{n+1}} = 1 + \frac pn + \alpha_n$$ and the series $\sum_{n=1}^{\infty} \alpha_n$ converges absolutely, ...
Yinuo An's user avatar
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1 vote
2 answers
78 views

Absolute convergence of series $\sum_{n=1}^{\infty }(\frac{\cos(n)}{\ln(n^{2n}+n^2)}+1-\cos(\frac{1}{n}))$

I have shown the convergence of the series by using Dirichlet's test to show that the first summand $\frac{\cos(n)}{\ln(n^{2n}+n^2)}$ converge and the comparison test to show that $1-\cos(\frac{1}{n})$...
nigatoni's user avatar
0 votes
0 answers
30 views

Weierstrass' M-test and convergence of series

I came across with the following proof about the convergence of Dirichlet L-series but I have troubles understanding it: Let $\delta >0$ and $f:\mathbb{Z}\rightarrow \mathbb{C}$ satisfy $|f(n)|\leq ...
Ishigami's user avatar
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Convergence in $𝐿^1$ and continuos CDF , implies converges in $𝐿^1$ for the corresponding binarized succession?

Let $X_n$ a succession of random variables with $\{x \in \mathbb{R}: \mathbb{P}(X_n=x)>0\} \subseteq[0,1]$ $\forall n$, and X a random variable with $\{x \in \mathbb{R}: \mathbb{P}(X=x)>0\} \...
Riccardo Cadei's user avatar
0 votes
0 answers
20 views

I want to know various ways to check convergence of infinite product

In complex analysis class I learned some condition that makes infinite product converge. Theorem : If $\sum |a_n|$ converges, then the infinite product $\prod(1+a_n)$ also converges. The proof is ...
SunnyMath's user avatar
  • 309
10 votes
2 answers
380 views

Associativity of infinite products

It is well-known that if $\sum_{n=1}^\infty a_n$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\...
Hilbert Jr.'s user avatar
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1 vote
0 answers
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Absolute Convergence of Fourier Series Proof

I am looking at the following theorem Let $f$ and $g$ be two piecewise continuous, periodic functions with the same period $p$ and with Fourier series $$\begin{align} \mathcal{F}[f]|_{t} = \sum_{k=-\...
Thomas Christopher Davies's user avatar
0 votes
2 answers
25 views

Absolute and conditional convergence of a non-alternating series

I tried all tests such as d'Alembert test, Cauchy test, Leibniz test,... but i could't determine convergence of this series: $$\sum_{n=2}^{\infty }\frac{(-1)^n}{\sqrt{n}+(-1)^n}$$ Can you help me!!!
kiyoshi_akira's user avatar
4 votes
1 answer
117 views

Convergence of series from inverse of Cauchy product

The Cauchy product of two real or complex infinite series $\sum_{n\in\mathbb{N}} a_n$ and $\sum_{n\in\mathbb{N}} b_n$ is defined as: $$ \forall n\in\mathbb{N}, c_n = \sum_{k=0}^n a_k b_{n-k} $$ ...
corindo's user avatar
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1 vote
1 answer
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How to Prove the Divergence of an Improper Integral Involving Absolute Value

I'm working on understanding the convergence properties of certain improper integrals and encountered the following integral: $$\int_{0}^{\infty} \left| \frac{\cos(x)}{\sqrt{x}} \right| \, dx$$ I ...
Matan Bitton's user avatar
2 votes
0 answers
45 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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Convergence of $\sum |a_n|$

I have question about convergence of $\sum |a_n|$, $1.$ If $\sum |a_n|$ converges then $\sum a_n^k$ converges of any $K \in N$, my reasoning was, an $\sum |a_n| $ converges which means $\exists N$ ...
Luckyian's user avatar
-1 votes
1 answer
63 views

The implications of an absolutely convergent Infinite Series [closed]

Does the convergence of $|a_n|$ imply the convergence of $\sqrt[4]{\frac{|a_n|}{n^4}}$ ? E.g. in R: $$\sum_{n=0}^{\infty} |a_n|\rightarrow a \in R \Rightarrow \sum_{n=0}^{\infty} \sqrt[4]{|\frac{...
Oblivious_Squid's user avatar
5 votes
0 answers
82 views

For which $\{a_k\}_{k=1}^\infty$ does $\sum_{k=1}^\infty \frac{1}{a_k} f(x+a_k)$ converge absolutely for almost every $x\in \Bbb R$?

Question: Let $f\in L^1(\Bbb R)$. For which increasing sequences $\{a_k\}_{k=1}^\infty$ of positive real numbers does $$\sum_{k=1}^\infty \frac{1}{a_k} f(x+a_k)$$ converge absolutely for almost every $...
stoic-santiago's user avatar
1 vote
0 answers
40 views

Find the limit of this series using the ratio test.

$a_n=\frac{c^n(n^2+3n+5)}{5^n(4n^n+3n+5)}$ where $c$ is a real constant Find the limit as $n \to \infty$ of this series. I’ve used the ratio test and written it in the form $\frac {a_{n+1}}{a_n} $ ...
Amy D's user avatar
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4 votes
0 answers
299 views

proof-read for : equivalence Every Cauchy sequence in R converges \iff Every absolutely convergent series in \R is convergent.

I just wrote a proof to show the following statement: Every Cauchy sequence in R converges iff Every absolutely convergent series in R is convergent. Please let me know what you think about the proof ...
isableisabel's user avatar
6 votes
0 answers
324 views

Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
Nomas2's user avatar
  • 667
0 votes
2 answers
257 views

Check the convergence of the series $\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$ and $\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$?

Check the convergence of the following series $$\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$$ and $$\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$$ My attempt: I tried Ratio test. I got \begin{align}...
Unknown x's user avatar
  • 849
3 votes
3 answers
195 views

Can we formally multiply out infinite products?

I came across Euler's proof that $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. One of the ingredients of the proof uses $$\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)=1-\...
Nomas2's user avatar
  • 667
1 vote
1 answer
55 views

If the absolute value of an infinite series is bounded above, then the infinite series absolutely converges?

I am attempting to follow this simple example found in Further Linear Algebra by Blyth & Robertson page 14. It deduces that a sequence of partial sums must be absolutely convergent when bounded ...
Jessica's user avatar
  • 13
0 votes
0 answers
60 views

Two hard problems of improper integrals

I have 2 problems that I have been stuck on: Check the absolute convergence and convergence of these improper integrals: a) $\int\limits_0^\infty x^p\sin(x^q)dx$ $(q\neq 0)$. b) $\int\limits_0^\infty\...
user avatar
2 votes
0 answers
96 views

Improper integral converges or not [duplicate]

Find all the values $\alpha \in(0,\infty)$ such that the improper integral $$\int\limits_0^\infty \frac{\Bbb dx}{1+x^{\alpha}\sin^2x}$$ is convergent. My attempt is to analyze the cases (i) $\alpha =1$...
user avatar
1 vote
1 answer
71 views

Convergent integral

Find all the value of $\alpha >0$ such that $\int\limits_0^\infty \dfrac{\sin x}{x^\alpha +\sin x}dx$ converges. My attempt is to check the convergence of $I_1= \int\limits_0^1 \dfrac{\sin x}{x^\...
user avatar
2 votes
0 answers
65 views

Integrability of the Jacobi Theta Function

Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that $$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem ...
Robert Abramovic's user avatar
-1 votes
1 answer
73 views

Absolute Convergence of a Series (maybe use ratio test) [closed]

Let $p:=\lim_{{k \to \infty}} k(1-|\frac{a_{k+1}}{a_k}|)$ Prove $p>1$ or $p=\infty \implies \sum_{n=1}^\infty a_n $ converge absolutely
John Frank's user avatar
0 votes
1 answer
171 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
0 votes
1 answer
86 views

does the series $\sum | \frac{(-1)^n}{\sqrt{n}} (1+ \frac{(-1)^n}{\sqrt{n}}) |$ converge?

Does the series $$S_n = \sum \bigg{|} \frac{(-1)^n}{\sqrt{n}} \left(1+ \frac{(-1)^n}{\sqrt{n}}\right) \bigg{|}$$ converge? I could arrive at $S_n = \sum \bigg{|} \left(\frac{1}{\sqrt{n}} + \frac{(-1)^...
Denis's user avatar
  • 441
0 votes
1 answer
92 views

The Expected Value of $g(X)$

I am studying "A First Course in Probability" by Sheldon Ross, and I have come across a problem with the following proof: Proposition 4.1 If $X$ is a discrete random variable that takes on ...
Arfin's user avatar
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1 vote
1 answer
65 views

Is $A(\Bbb T) \subset C(\Bbb T)$?

My instructor defined the space $A(\Bbb T)$ as $$A(\Bbb T) := \left\{f\in L^1(\Bbb T): \sum_{n=-\infty}^\infty |\hat f(n)| < \infty\right\}$$ and wrote $A(\Bbb T) \subset C(\Bbb T)$, where $C(\Bbb ...
stoic-santiago's user avatar
0 votes
1 answer
64 views

At which value of K, the sum converges

I have a infinite sum that looks like the following: $$ \sum_{k = 0}^{\infty} \left(\frac{1}{1 + k/A}\right)^a \frac{(-i B)^k}{k!} $$ Here, A is a positive real number and $a$ is a positive integer. ...
CfourPiO's user avatar
  • 109
2 votes
2 answers
211 views

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\...
ANT Learner's user avatar
0 votes
0 answers
18 views

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\...
Swetha Rao's user avatar
0 votes
1 answer
83 views

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 ...
entropy's user avatar
  • 147
1 vote
0 answers
35 views

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-...
Bruno B's user avatar
  • 5,859
1 vote
1 answer
134 views

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. ...
Eparoh's user avatar
  • 1,279
2 votes
3 answers
161 views

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. $$\...
Dylan Levine's user avatar
  • 1,688
0 votes
2 answers
47 views

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{...
Dargscisyhp's user avatar
0 votes
0 answers
68 views

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
5 votes
1 answer
94 views

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) \; \; \; \...
givo's user avatar
  • 51
4 votes
0 answers
75 views

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 ...
Megatron's user avatar
  • 111
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
0 votes
0 answers
22 views

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 ...
ygh's user avatar
  • 121
2 votes
0 answers
129 views

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 ...
nomadicmathematician's user avatar
1 vote
1 answer
190 views

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)\...
Marvin Ray Burns's user avatar
0 votes
2 answers
68 views

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'...
langmai's user avatar
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