Questions tagged [absolute-convergence]

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

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1answer
37 views

“Normal convergence implies uniform convergence and absolute convergence”. Attempt to prove it via Weierstrass M-test. What is missing?

Let us start with a given function $f_n$ defined on a certain domain $I$ and s.t. $f_n:I\mapsto\text{ some normed vector space}$. We know that: $$\sum_{n=0}^\infty \|f_n\| := \sum_{n=0}^\infty \sup_I |...
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1answer
26 views

Formula for the “standard” series $\sum_{k=0}^\infty{\frac{\lambda^k}{k!}}$

Trying to remember if there is a general formula for this series: $$ \sum_{k=0}^\infty{\frac{\lambda^k}{k!}} = \sum_{k=0}^\infty{\frac{e^{ln(\lambda)k}}{k!}} $$ Alternatively is there a formula for ...
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3answers
20 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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2answers
32 views

Absolute and conditional convergence of series proof

Assume $ \sum_{n=1}^{\infty}a_{n} $ is absolutely convergent. and assume $ \sum_{n=1}^{\infty}b_{n}$ is conditionally convergent. What can we say for sure about $ \sum_{n=1}^{\infty}\left(a_{n}+b_{n}\...
2
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0answers
47 views

Banach algebra structure of $\ell^p$ sequence space

Long intro: questions after examples. The space of absolutely convergent sequences $\ell^1 = \{ \vec{a} = (a_1, a_2, a_3, ...) \in \Bbb{R}^\Bbb{N}: \sum_{n} |a_n| < \infty \}$ is closed under ...
1
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2answers
69 views

Convergence of a series: question

I know $$ \lim_{n \to \infty}\left\{\,% {2\cos\left({\left[k - 1\right]\pi \over n-3}\right) - 2\cos\left(k\pi \over n + 1\right)}\,\right\} = 0, $$ for $k \in \mathbb{N}$. $$ \mbox{Can I conclude}\...
5
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1answer
157 views

What can be said about the sum of the series?

Let $\{a_n \}_{n \geq 1}$ be a sequence of non-zero integers satisfying I. $|a_n| \lt |a_{n+1}|,$ for all $ n \geq 1$ II. $a_n$ divides $a_{n+1},$ for all $n \geq 1$ and III. every integer is a ...
3
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1answer
69 views

Show that $\sum\frac{(-1)^{n+1}} {{n}^r} \sum\frac{(-1)^{n+1}} {{n}^s} $ by Abel's rule forms a series that doesn't converge when r+s=1.

It is a similar problem to that in Show that the series $\frac{1} {\sqrt{1}} -\frac{1} {\sqrt{2}} +\frac{1} {\sqrt{3}} +\dots$ converges, and its square (formed by Abel's rule) doesn't.. It ...
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2answers
131 views

Convergence of $\sum_{n=1}^\infty 2^n\sin\frac{1}{3^nz}$

The problem is: prove $\sum_{n=1}^\infty 2^n\sin\frac{1}{3^nz}$ converges absolutely for all $z\neq 0$, but does not converge uniformly near $z=0$. Proof: for all $z\neq 0$ $$\left|2^n\sin\frac{1}{3^...
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1answer
51 views

Prove that $\ \prod_{n=1}^{\infty} \{(1-\frac{z}{n})^{nk} \exp(\sum_{m=1}^{k+1}\frac{n^{k-m}z^m}{m}) \}$ converges absolutely.

Prove that $\forall z,\ \prod_{n=1}^{\infty} \{(1-\frac{z}{n})^{nk} \exp(\sum_{m=1}^{k+1}\frac{n^{k-m}z^m}{m}) \}$ where $k$ is a positive integer, converges absolutely. It seems to me $(1-\frac{z}{n})...
2
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1answer
38 views

Beginner Fourier Problem and absolute convergence

presumably as preparation for Fourier Analysis, I was given the following exercises: Let $\sum_{n=1}^{\infty} a_{n}$ be a series which is absolute convergent and for $x\in\mathbb{R}$ $$ f(x)=\sum_{n=...
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3answers
92 views

Proof of Absolute Convergence of $\sum\limits_{n=1}^{\infty} (-1)^{n-1}\tan\left(\frac{1}{n\sqrt{n}}\right)$

$$\sum\limits_{n=1}^{\infty} (-1)^{n-1}\tan\left(\frac{1}{n\sqrt{n}}\right)$$ My Attempt: To prove absolute convergence, we must consider $\sum\limits_{n=1}^{\infty}a_n$ and $\sum\limits_{n=1}^{\infty}...
1
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0answers
41 views

Convergence of $\sum_n e^{-a_nz}$

Let $a_n > 1, n>0$ be an increasing unbounded sequence of real numbers. Is it true that $\sum_n e^{-a_nz}$ converges for all $z>0$? If not, then is it at least true that $\sum_n e^{-a_nz}(\...
0
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1answer
34 views

Convergence and absolute convergence of an infinite product of terms in $(0,1]$.

Let $x_i\in(0,1]$ for all $i$. Are the following true, and if so is there a easy proof or citation. $\prod_{i=1}^\infty x_i = e^{\sum_{i=1}^\infty \log x_i}$ always holds (if the sum diverges to $-\...
1
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0answers
19 views

Termwise differentiation of a Fourier Series

I’m asking for clarification regarding the proof of a theorem characterizing the solutions to the steady state heat equation on the 2D unit disc. This proof appeared in Stein’s analysis textbook from ...
2
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1answer
56 views

Taylor series expansion rearrangement

From the Taylor series expansion of $\sqrt{x}$ provided in Algorithms for approximating $\sqrt{2}$ I got Taylor series expansion of $x^{s}$, $0<s<1$ and then collecting the coefficients of $x^{i}...
1
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1answer
50 views

Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely

Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely. I get why the statement is correct (because it means that the convergence of the series ...
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0answers
56 views

Show that the series $\sum\limits_{n\ge1}a_nX_n$ converges absolutely a.s. for some constants $a_n\ne0$.

Let$\{X_n\}_{n\ge1}$ be an arbitrary sequence of random variables. Show that the series $\sum\limits_{n\ge1}a_nX_n$ converges absolutely a.s. for some constants $a_n\ne0$. My attempt at a solution: ...
1
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1answer
63 views

Ratio Test intuitive Idea

My attempt to understand the proof is as follows: If $R=\lim\sup\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|<1$ implies that for $\epsilon$ such that $0<\epsilon<1-R$ we can find an $...
0
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1answer
30 views

Absolute convergence of a series with integral inside

Im having trouble solving the absolute convergence of this series. None of the common tests seem to work and so far couldn´t find any function to compare it to: $\sum_{n=1}^\infty (-1)^{n} \int_{n}^{n+...
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1answer
227 views

How can I prove that this function is bounded?

The binomial series of $(1-x^n)^{\frac{1}{n}}$, where $n$ is a positive integer, converges absolutely to $(1-x^n)^{\frac{1}{n}}$ for $x\in[0,1]$. The binomial series expansion is $$(1-x^n)^{\frac{1}{n}...
1
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1answer
21 views

Root test for complex series and cancelling powers with absolute values

The root test for convergence of a complex power series is given as $$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = L$$ If $a_n = \frac1{(1+i)^n}$ then I read that when applying the root ...
1
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2answers
57 views

Showing a series converges absolutely

The goal is to prove that if $|\frac{c_{n+1}}{c_n}|\leq1+\frac{a}{n}$, where $a<-1$ and $a$ does not depend on $n$, then the series $\sum_{n=1}^\infty c_n$ converges absolutely. My idea: to have ...
1
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1answer
28 views

Missing Step in Proof that Completeness implies that all Absolutely Convergent Series Converge

I am trying to prove that if $X$ is a Banach space, then every absolutely convergent series in $X$ converges in $X$. My current proof is below but I realize that, in the first paragraph, the first &...
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2answers
46 views

If $T$ is a bounded linear map and $\sum x_n$ is an absolutely convergent series, then $T(\sum x_n) = \sum T(x_n)$

Is the following true? If so, how to prove it? If $T:X \to Y$ is a bounded linear map between the Banach space $X$ and the normed vector space $Y$ and $\sum x_n$ is an absolutely convergent series, ...
0
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2answers
13 views

Absolute convergence and continuty of $\sum_{n=1}^\infty \sin(\frac{x}{n^4})\cos(nx)$

Question: Prove that the series $$\sum_{n=1}^\infty \sin(\frac{x}{n^4})\cos(nx)$$ is converges absolutely, and is continuous on $\mathbb{R}$. Attempt: I can readily see that the series converges ...
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2answers
17 views

Absolute-convergence of infinite series

Precondition; limsup [n→∞] (|a_n|・|x-c|^n)^(1/n)=0 for all x. Problem; Prove the fact that if the above precondition works, \sum_{n=0}^∞ |a_n・(x-c)^n|<∞ for all x. I can't understand why \sum_{n=0}^...
0
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1answer
61 views

proof that $\lim \sup (a_n · b_n) = \lim \sup a_n · \lim \sup b_n $

With $a_n, b_n \geq 0$ or non-negative for all $n$ in $\mathbb N$. Proof that for $n \to \infty $ $\lim \sup (a_n · b_n) = \lim \sup a_n · \lim \sup b_n $ , if $a_n$ and $b_n $converge. My start ...
1
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2answers
65 views

Infinite series of sequences

Let S be the set of sequences whose series converge absolutely. We define 2 norms on S: $$\| \{ a_n \}_{n=0}^{ \infty } \|_1 = \sum_{n=0}^\infty | a_n |$$ and, $$\| \{ a_n \}_{n=0}^\infty \|_{\sup} = \...
0
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2answers
22 views

Convergence of functional series $\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}$

Find convergence area and absolute convergence area of the following series: $$\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}$$ I wrote it as $$\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}=\...
1
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1answer
45 views

How the series is convergent and absolute convergent

$$\sum_{n=1}^\infty (-1)^{n-1} \frac{\sin(nx) + \cos(nx)}{n^{3/2}}$$ I tried to prove convergence by Leibnitz theorem but can't prove absolute convergent please give me some hint about this.
0
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2answers
35 views

Convergence of a series involving $e^{in\phi}$

Consider the following series: $\displaystyle\sum_{n=1}^{\infty} \frac{(n+1)e^{in\phi}}{n^2}, \phi \in \mathbb{R}, \phi ≠ 2\pi k$ for $k \in \mathbb{Z}$ Then: $\displaystyle\sum_{n=1}^{\infty} \...
2
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2answers
91 views

Prove the improper integral is absolutely convergent

$$\int_0^{\pi/2} \sin(\sec(x))dx$$ This is what I come up with: Let $ \sec x=\frac{1}{t}, \cos x=\frac{1}{t} $, $x=\arccos\frac{1}{t}$, so $dx = -\frac{1}{\sqrt{1-\frac{1}{t^2}}}\cdot \left(-\frac{1}{...
0
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2answers
66 views

Convergence of power series with factorial ratio coefficients

$$\sum _{n=0}^{\infty }\:2^{2n}\cdot \frac{\left(n!\right)^2}{\left(2n\right)!}x^n$$ Using ratio test we can see that radius of convergence is $R = 1$. Though I'm not sure how to find the exact ...
3
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1answer
75 views

Does $\sum_{n=1}^{\infty} a_n$ is absolutely convergent $\Rightarrow$ $\sum_{n=1}^{\infty} a_n\sin(nx)$ is absolutely and uniformly convergent?

I'm proving that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent $\Rightarrow$ $\sum_{n=1}^{\infty} a_n\sin(nx)$ is absolutely and uniformly convergent. I defined for a set $\mathbb{D} \subset \...
0
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1answer
33 views

Is the series comparison criterion for convergence valid for absolute convergence?

I'm studying the convergence and absolute convergence of the series of functions defined by the sequence of functions: \begin{equation*} f_n: \mathbb{R} \to \mathbb{R}, \end{equation*} \begin{...
0
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4answers
110 views

How is $\displaystyle\sum_{n=1}^{\infty} \frac{n^43^n}{n!}$ absolutely convergent?

For $\displaystyle\sum_{n=1}^{\infty} \frac{n^43^n}{n!}$ we use the ratio test. By $\lim_{n\to\infty}$ my end result is that it goes to $0$ and is convergent. I do not know how to prove it is ...
2
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1answer
51 views

Absolute convergence of $\int_{0}^{1} \left(x\cos\left(\frac{\pi}{x}\right)\right)' dx$

$\int_{0}^{1} \left(x\cos\left(\frac{\pi}{x}\right)\right)' dx$ converges. Does it absolutely converges? I found easy to show that it converges (by a direct computation or the subtraction of limits). ...
1
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2answers
24 views

Convergence of Taylor Series (as part of finding the region of Conv for a Laurent series)

I'm given a Laurent series problem and to find the largest region of convergence, I need to fine $R_1$ and $R_2$. The Laurent series is $$z^3 - \frac{z}{3!} + \sum_{k\geqslant2} \frac{(-1)^{k}}{(2k+1)...
0
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1answer
35 views

Finding the largest region on which Laurent series converges

I need to find the largest region on which $$f(z) = z^4 sin(1/z)$$ defined on $$\mathbb{C}/{0}$$ converges. So for the Laurent series, I got $$z^3 - \frac{z}{3!} + \sum_{k\geqslant2}\frac{-1^{2k+1}}{(...
0
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1answer
34 views

Prove that $\sum_{n=1}^\infty |\frac{1}{a_{n+1}} - \frac{1}{a_n}| \space converge \Longleftrightarrow \sum_{n=1}^\infty |a_{n+1} - a_n|$ converge

let $(a_n)$ be a sequence, $\forall n, a_n \ne 0$, and $a_n \to a \ne 0$ so assuming $a_n \ne a_{n+1}$ I have already proven that: $$\lim_{n\to \infty} \frac{|a_{n+1} - a_n|}{|\frac{1}{a_{n+1}} - \...
1
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0answers
48 views

Convergence of double sum and its rearrangement

Let $A=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij},a_{ij}\geq 0$ and $$\phi:\mathbb N \to\mathbb N\times\mathbb N$$ be any bijection. Now $B=\sum_{k=1}^{\infty}a_{\phi(k)}$. How $A$ and $B$ are ...
0
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3answers
78 views

For which $x$ is $ \sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$ (absolutely) convergent?

Consider the power series: $$ \sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$$ with a, b >0 : a) for which x is this series absolute convergent, b) for which x is this series conditionally convergent, ...
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1answer
31 views

convergence of an improper interval with two functions that are battling each other to see if the integral of the product converges

Boiled down, I have two functions of x and I need to know if the indefinite integral of their product converges. In particular, if: $$y=x\exp{-x^2}$$ then how can I show (if its true) that the ...
2
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1answer
34 views

How to prove this series is absolute convergent?

Assume $X$ is a Hilbert space, $\{e_n\}_{ n=1}^{\infty}$ is an orthogonal set of $X$. Let $x=∑α_ne_n$, $y=∑β_ne_n$. Prove $(\mathrm{x}, \mathrm{y})=\sum_{\mathrm{n}=1}^{\infty} \alpha_{\mathrm{n}} \...
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1answer
64 views

proving if $\sum_{n=2}^{\infty} \frac{n\cdot \cos(2n)}{n^2 + 11} + \frac{\cos(\pi n)}{\ln(n)\cdot \ln(n^n + n)}$ converge conditionally or absolutely

I have already proven that $$\sum_{n=2}^{\infty}\frac{\cos(\pi n)}{\ln(n)\cdot \ln(n^n + n)}$$ conditionally converge and $$\sum_{n=2}^{\infty} \frac{n\cdot \cos(2n)}{n^2 + 11}$$ converge, hence $$\...
0
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1answer
63 views

If the series is absolutely convergent, then it is also conditionally convergent.

Let $\sum_{n=m}^{\infty}a_{n}$ be a formal series of real numbers. If the series is absolutely convergent, then it is also conditionally convergent. Furthermore, in this case we have the triangle ...
0
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0answers
41 views

Heaviside Function: as a sequence/series, converging absolutely, left hand and right hand limits, continuity, and monotonicity

I have a question about the definition of the Heaviside function $H(x)$ as follows: $$H(x)= \begin{cases} 0 & x < 0, \\ 1 & x\geq 0. \end{cases} $$ Let $(...
0
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1answer
16 views

How to find the values of the entries of partial sums of a conditionally convergent series?

It is given that the series $\sum_{n=1}^\infty a_n$ is convergent, but not absolutely convergent and $\sum_{n=1}^\infty a_n=0$. Denote by $s_k$ the partial sum $\sum_{n=1}^k a_n,k=1,2,.....$. Then $...
1
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1answer
24 views

Convergence of a metric given by a series

Let $\{{0, 1}\}^{\omega}$ be the set of all binary sequences, i.e., sequences of zeros and ones. It is pretty straightforward to verify that $$d(x, \, y) := \sum_{n\,=\,1}^{\infty} \frac{\mid{\,x_n - ...

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