Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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1
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2answers
193 views

Disprove or prove $\sum_{n\ge1} \frac{\vert{\sin n}\vert}{n} $ is divergent. [duplicate]

I want to prove or disprove $\displaystyle\sum_{n\ge1} \frac{{\sin n}}{n}$ is absolutely convergent. I can prove $\displaystyle \sum_{n\ge1} \frac{{\sin n}}{n}$ is convergent by dirichlet's test (Can ...
3
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1answer
61 views

Two problems on real number series

Consider the series: $$a) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\sin\frac{1}{2k})}$$ $$b) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\tan\frac{3}{2k})}$$ Showing that these two are ...
3
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1answer
565 views

Using Dirichlet's test

I am trying to find out if the following series converges or diverges. If it converges I then want to find out if it converges absolutely or conditionally. $$\sum _{n=1}^{\infty}\frac{\sin(n)}{\sqrt ...
3
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1answer
66 views

Four (simple) questions on real number series

I'm trying some exercises on real number series, in which I have to see if the series are convergent or not: $$a) \sum_{n=1}^\infty \ (-1)^n \frac{2n+1}{3^n}$$ $$b) \sum_{n=0}^\infty \ (-1)^{n+1} \...
3
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1answer
260 views

If the radius of convergence of the power series $a$ is positive then it “reciprocal” power series have positive radius of convergence

Im trying to solve this exercise from the book Analysis I of Amann and Escher (page 216, exercise 9). Let $a=\sum a_k X^k\in\Bbb C[\![X]\!]$ with $a_0=1$. (a) Show that there is some $b\in\Bbb ...
1
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0answers
136 views

Kummer's transformation applied to a series involving the Möbius function: were rights my deductions and how get an idea of the improvement?

After I've read Kummer's recipe to get the acceleration of a series, I want to do an example related with the Möbius function $\mu(n)$. Question. I present my calculations, please A) I would ...
0
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1answer
31 views

Numerical series in terms of alternating sign

Given the number series: $\begin{aligned}\sum_{n=1}^{+\infty}\end{aligned} (-1)^n \log\left(\frac{2}{\pi}\,\arctan\sqrt{n}\right) $ since both the criterion of Leibniz, is the absolute convergence ...
2
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4answers
119 views

Does each term in an infinite series have to be greater than the infinite sum after it? [closed]

I remember a teacher telling me that the absolute value of any term in a sequence will exceed some forms of sum of the infinitely many terms after it. I do not remember if it was 'the absolute value ...
2
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1answer
107 views

A definition of the functional $f\to\sum_{n=1}^\infty\mu(n)f \left( \frac{1}{n} \right) $, involving good functions $f$ and the Möbius function

I would like to create a definition, is a conditional definition since the behaviour of arithmetic functions like to the Möbius function $\mu(n)$ depends on the veracity of the Riemann Hypothesis, ...
2
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2answers
134 views

Geometric series convergence and upper bound of its absolute value

If I know the following: $1/4 \le |X_n| \le 1/2$ for all $n\ge 1$, then I must prove that $$\sum_{n=1}^\infty (X_n)^n$$ converges and that $$ \left|\sum_{n=1}^\infty (X_n)^n\right|\le 1$$ I used ...
-1
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2answers
41 views

Chck if the series $\sum_{n\ge1}{\frac{i^n}{n}}$ absolutely convergent, semi convergent or divergent?

Is the next series absolutely convergent, semi convergent or divergent? $$\sum_{n\ge1}{\frac{i^n}{n}}$$ where $i$ is the complex number such that $i^2=-1$. I don't know that I can use Leibniz ...
2
votes
1answer
89 views

Absolute convergence of a series and parameters

I've been taught that if a series is absolutely convergent, then it is convergent. I have a question, because my series has got a parameter within it. Let's call it b. If a series is absolutely ...
4
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1answer
171 views

Prob. 11 (d) in Baby Rudin: Given $a_n > 0$, is this condition also sufficient for divergence of $\sum \frac{a_n}{1+na_n}$?

Here's Prob. 11 (d), Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $a_n > 0$ and that the series $\sum a_n$ is divergent. Then what can be said ...
0
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2answers
69 views

Question related to convergence of series $\sum_{n=1}^\infty (-1)^n\frac {x^2+n} {n^2}$.

We have the series $\displaystyle\sum_{n=1}^\infty (-1)^n\frac {x^2+n} {n^2}$. Which test ensure that the series convergence for all real value of $x$ and how can we confirm that this series does not ...
0
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1answer
52 views

$\sum_{n \neq 0} {\dfrac{\sin(n\vartheta)}{n}}$ does not converge absolutely, when $0 < \vartheta < \pi$.

I want to show that the series $\sum_{n \neq 0} {\frac{\sin(n\theta)}{n}}$ does not converge absolutely when $0 < \theta < \pi$. When $\theta$ is of the form $\frac{a}{b}\pi$ with $a,b \in \...
0
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1answer
51 views

Absolute convergence for $\sum_1^n(f(x))^n$ where $f\in C([0,1])$ and $\|f\|_\infty=1$

I am studying for a final exam I have tomorrow and am having trouble solving this practice problem. Let $f \in C [0, 1]$, such that $\|f\|_\infty =1$, where $\| \cdot \|_\infty $ denotes the sup-...
4
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2answers
145 views

Question on a Proof of Rearrangements for Absolutely Converging Double Series

In Appendix B of Jameson's The Prime Number Theorem, the author gives a proof of the assertion that given real numbers $\left\{a_{j,k}\right\}_{j,k\ge 1},$ if $$\sum_{j=1}^\infty \sum_{k=1}^\infty \...
1
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1answer
55 views

I'm having trouble with with improper integral convergence

i am having though time trying to find if $\int_2^\infty \frac{\cos(x^2)}{x\ln(x)}dx$ converges. does it converge or absolutely converging? i am sorry that i am not writing with the correct symbols ...
0
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2answers
419 views

determine whether the series is absolutely convergent, conditionally convergent, or divergent [closed]

does the series $((-2)^n)/(n^2)$ from $n=1$ to infinity converge or diverge? Is the ratio test applied?
1
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1answer
88 views

About absolute convergent series in Banach space and its bounded linear functional.

Let $\{x_n\}$ be a sequence in Banach space $X$ such that $\sum_{n=1}^{\infty} |f(x_n)|< \infty$ for all bounded linear functional $f \in X'$. Show that there exists $M\geq 0$ such that $\sum_{n=...
1
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1answer
210 views

double summation absolute convergence

I'm trying to prove or disprove the absolute convergence of the sum $\sum_\limits{k,l=1}^{\infty}\frac{k-l}{k^4+l^4} $. Every series that I found that bounds the absolute values of my original ...
2
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1answer
31 views

Not seeing the justification for an absolutely summable series satisfying an inequation.

I'm reading in Dieudonne: In order that a denumerable family $(x_\alpha)_{\alpha\in A}$ of elements of a Banach space $E$ be absolutely summable, a necessary and sufficient condition is that ...
0
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1answer
188 views

double summation binomial coefficient

I have the following sum to evaluate: $ \sum_\limits{l,k=0}^{\infty} \binom{l}{k} (-1)^kr^{k-2l} $ . I feel like I first have to establish absolute convergence for a certain range of values of $|r|$...
1
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0answers
24 views

On the integral representation of the function $b(s)$ defining $b(n)=\operatorname{Bernoulli}_{2n}$, and Riemann's trick

I don't know if this approach was in the literature, I would like to know some expression for $\Re s>1$ of the product $$\zeta(s)b(s),$$ where $\zeta(s)$ is the Riemann Zeta function, and $b(s)$ is ...
2
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2answers
243 views

Example of a series of functions that converges absolutely but does not converges uniformly?

Can someone suggest an example of a series of functions that converges absolutely but does not converges uniformly?
2
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2answers
387 views

Eisenstein series converge absolutely for $k\geq 2$

I am looking at Eistenstein series on modular forms: https://en.wikipedia.org/wiki/Eisenstein_series The page claims that the series converges absolutely to a holomorphic function of τ when $k\geq 2$....
1
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0answers
127 views

Prove absolute and uniform convergence on compact subsets of $\mathbb{C}$

Consider the series $$\sum_{n=-\infty}^{\infty} e^{\pi i [n^2 \tau + 2n z]},$$ where Im$(\tau) >0$ and $z \in \mathbb{C}$. Prove that this series converges absolutely and uniformly on compact ...
2
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1answer
228 views

Alternating series with sin

I have this alternating series: $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n+2\sin n}$$. Leibniz test and the absolute convergence didn't work. Neither did the divergence test. When showing that $a_n=\dfrac{...
1
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1answer
70 views

Is this series convergent

The question is regarding how to solve a general set of series. If a series whose nth terms goes like this a= (-1)^n/ (n^3+2n) , how should I find whether the series is convergent or not. Since the ...
2
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1answer
40 views

Convergence of $\sum_{n=0}^\infty \left (\frac{(-1)^n}{n+1}+\frac{(-1)^{n+1}}{n+2} \right )$

What would you say about this series about convergence and absolute convergence? $$\sum_{n=0}^\infty \left (\frac{(-1)^n}{n+1}+\frac{(-1)^{n+1}}{n+2} \right )$$ In use with: $$\sum_{n=1}^∞ \frac{1}{...
-3
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1answer
36 views

Testing for divergence and convergence [closed]

$$\sum_{k=1}^n (1/2)$$ Why/how does this series diverge? Why doesn't it converge?
0
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1answer
24 views

finding the limit of a series and testing for convergence.

$\lim \limits_{k \to \infty}\cos(kπ) + \sin(k(π/2))$ = undefined, so the series diverges. How is this undefined? isn't undefined when you get 0/0? How would I go about finding out what happens as ...
0
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2answers
55 views

Can we argue that as $\sum^\infty _{n=m} a_n$ is $\sum^\infty _{n=m}|(-1)^n a_n|$, $\sum^\infty _{n=m}(-1)^n a_n$ is convergent?

I want to prove Let $(a_n)^\infty_{n=m} $ be a sequence of real numbers which are non negative and decreasing, thus $a_n\geq 0$ and $a_n\geq a_{n+1}$ for evere $n \geq m.$ Then the series $\sum^\...
0
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1answer
50 views

Does this double-sum converge?

This is a very simple question, but I don't know if this is correct or not. Is the following sum $$\displaystyle \sum_{\substack{\ \ k,l \in \mathbb{Z} \backslash \{0\}} \\ {\ \ \ \ \ \ \ k,l \neq 0}...
1
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1answer
75 views

Prove that if $\sum_{n=1}^\infty|a_n| < \infty$, then $\{a_n\} \in \ell^2$

Prove that if $\sum_{n=1}^\infty|a_n| < \infty$, then $\{a_n\} \in \ell^2$ I'm having a little trouble getting started with this proof. Here's what I think we need to show: $$\sum_{n=1}^\infty |...
4
votes
1answer
81 views

Show that $\sum_{n = 1}^{\infty} \frac{1}{z + n} + \frac{1}{z-n}$ is absolutely convergent for all $z \in \mathbb{H}$

I need to show that the series $$ \sum_{n = 1}^{\infty} \frac{1}{z + n} + \frac{1}{z-n} $$ is absolutely convergent for all $z$ in the complex upper half plane $\mathbb{H} = \{ z \in \mathbb{C} : \Im(...
1
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1answer
95 views

Absolute convergence of the Fourier series

Given a Fourier series: $$f(x) = \sum_{n=1}^\infty (a_n \sin(nx)+b_n \cos(nx))$$ Show it is absolutely convergent if $$\sum^\infty_{n=1}(|a_n|+|b_n|)$$ is finite. My attempt to show it is as ...
0
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3answers
48 views

How do we find the sum of this series $\sum [a+(n-1)d] b r^{n-1}$?

Let $a, b, d, r$ be real numbers such that $d \neq 0$ and $r \neq 0$. Let $$s_n \colon= [a+ (n-1)d] b r^{n-1}$$ for $n=1, 2, 3, \ldots$. Then how do we find $$\sum_{n=1}^N s_n$$ for $N = 1, 2, 3, \...
1
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1answer
21 views

Absolute convergence of $\sum_{n\geq 0} (f_n(x)+g_n(x))$ when $\sum_{n\geq 0} g_n(x)$ or $\sum_{n\geq 0} f_n(x)$ or both converge only conditionally

Considering two function series i know that $$\sum_{n\geq 0} f_n(x) \mathrm{\,\, converges \,\,\, absolutely \,\,\, and } \sum_{n\geq 0} g_n(x) \mathrm{\,\, converges \,\,\, absolutely} \implies \sum_{...
2
votes
1answer
305 views

Prob. 13, Chap. 3 in Baby Rudin: The Cauchy product of two absolutely convergent series converges absolutely

Here's Prob. 13, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Prove that the Cauchy product of two absolutely convergent series converges absolutely. My ...
1
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1answer
78 views

Prob. 12, Chap. 3 in Baby Rudin: Some results involving the remainder of a convergent series of positive term series

Here is Prob. 12, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $a_n > 0$ and $\sum a_n$ converges. Put $$ r_n = \sum_{m=n}^\infty a_m.$$ ...
2
votes
2answers
275 views

Prob. 7, Chap. 3 in Baby Rudin: If $a_n \geq 0$, then how does convergence of $\sum a_n$ imply convergence of $\sum \frac{\sqrt{a_n}}{n}$? [duplicate]

Here's Prob. 7 in the Exercises of Chapter 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Prove that the convergence of $\sum a_n$ implies the convergence of $$ \...
1
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1answer
128 views

Prove that if the power series $\sum_{n=0}^\infty a_nx^n$ converges for $x = x_0$, then it converges absolutely for all $x$ such that $|x| <|x_0|$

Prove that if the power series $\displaystyle \sum_{n=0}^\infty a_nx^n$ converges for $x = x_0$, then it converges absolutely for all $x$ such that $|x| <|x_0|$. I am supposed to use the following ...
1
vote
1answer
50 views

Absolute Continuity for 0 Radon Nikodym derivative set

Let $\nu$ and $\mu$ be positive $\sigma$-finite measures on $(S,\Sigma)$ with $\nu \ll \mu$ and let $h = \frac{d\nu}{d\mu}$. I want to show that \begin{align} \mu\big(\{h=0\}\big)=0 \iff \mu \ll \nu....
2
votes
2answers
219 views

Theorem 3.55 in Baby Rudin: How to make sense of the proof?

Here's Theorem 3.55 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement ...
1
vote
1answer
50 views

Convergent Series and Rearrangements

Given a convergent positive series, I have to prove that $$\sum^{\infty}_{n=1} a_n=\sum^{\infty}_{k=0}a_{2k+1} + \sum^{\infty}_{n=1} a_{2k}$$ which means that the sum of odd terms and the even terms ...
0
votes
1answer
48 views

Asymptotic behaviour of absolute convergent series

$\newcommand{\nn}{\mathbb N}\newcommand{\abs}[1]{\left| #1 \right|}\newcommand{\d}{\,\mathrm d}$When I was thinking about a problem I thought the following conjecture to be true. $$\sum_{n \in \nn} ...
0
votes
1answer
82 views

Proving a space is a Banach space.

Let $x_1, x_2, \ldots$ be a Cauchy sequence in a normed vector space $X$. Suppose that any absolutely convergent series converges. Then: prove that $x_1, x_2, \ldots$ converges. My idea was: set $y_n ...
7
votes
4answers
171 views

Find an example of series which converges only absolutely on $\mathbb Q$

I am currently working on the completeness of metric spaces, so I studied the following theorem: If $E$ is a Banach space then any absolutely convergent series is convergent. Since $\mathbb Q$ is ...
2
votes
1answer
574 views

Normed space where all absolutely convergent series converge is Banach [duplicate]

Let $A$ be a normed space where every absolutely convergent series converges in $A$. How do I prove that $A$ is Banach? Let $\sum_{n=1}^\infty x_n$ be an absolutely convergent series in $A$, then ...