Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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2
votes
1answer
28 views

Convergence and Divergence in $2^{-\ln n}$

The series that I need to check is $2^{-\ln n}$ and $3^{-\ln n}$. For the second case, after doing the preliminary test (which gave me zero so that I may proceed further), I tried to solve it by ...
0
votes
1answer
28 views

Show that the next double power series is convergent

Show that $$\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}[k|\beta +1]_n\frac{|t|^k}{k!}\frac{|x|^n}{n!}$$ is convergent $\forall t \in \mathbb{R}$, and $x, \beta \in \mathbb{R}$ such as $|\beta x|<1$, ...
4
votes
2answers
103 views

Convergence of $\sum_{k=1}^{\infty}\frac{\cos(\theta k)}{\sqrt{k}}$

Say if the following series $$ \sum_{k=1}^{\infty} \frac{\cos(\theta k)}{\sqrt{k}} $$ for $θ \in \mathbb{R}$ is convergent. Is it absolutely convergent? I don't know how to approach this problem. ...
0
votes
1answer
48 views

Sum of the Riemann-Zeta function at the integers

If you consider the series of reciprocal positive integers (the harmonic series), the sum diverges. However, some subsets of this series, such as the reciprocal cubes, will sum to something finite. I ...
0
votes
1answer
28 views

Absolute convergence of $\sum_{k=0}^{+\infty}a_{n}(x+2)^n$

Define the series $$\sum_{n=0}^{+\infty}a_{n}(x+2)^n.$$ All we know is that it converges in x=4. Why can't I say it converges absolutely in x=4?
1
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2answers
54 views

If $\sum a_n$ converges absolutely, then so does $\sum f(a_n).$

Hi I am stuck on this problem. (please dont give the solution I just need some help to formalism my solution) Let f: R to R be differentiable, with continuous derivatives and f(0) = 0 If $\sum{a_{n}}$...
0
votes
1answer
22 views

series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...
0
votes
1answer
46 views

Definitions of sin and cos using the exponent

I follows these steps: Define $e^z:=\sum_{k=0}^{\infty}\frac{z^k}{k!}$. Show thatthe series is absolutely convergent. Define $\sin(z):=\frac{e^{iz}-e^{-iz}}{2i}$, and $\cos(z):=\frac{e^{iz}+e^{-iz}}{...
0
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0answers
16 views

Absolute convergence of series variant of the geometric series

So I want to prove whether the following series converges absolutely or not: $$\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)q^n$$ where $ q \in \mathbb{C}, \mid q\mid<1.$ My attempt was: $$\frac{1}{2}...
0
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0answers
33 views

Prove that the sum of a series is differentiable

Prove that the series $$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$ is absolutely and uniformly convergent on $\mathbb{R}$. If the sum of the series is denoted by $f(x)$ prove that $f$ is differentiable at ...
2
votes
3answers
45 views

Convergence of the series below

$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$ I did: $$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$ $$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$ So diverges by the Ratio ...
2
votes
1answer
23 views

Grouping the Summation

Let $a_n \in \mathbb{C}$ and consider $\sum a_n$ and grouping as $\sum (a_n + a_{n+1})$. Under what assumptions we can claim absolute convergence of grouped sum implies convergence of the original sum?...
3
votes
1answer
44 views

Is $\ell^1$ complete with this norm?

For $x \in \ell^1$ we set $\Vert x\Vert = \sup\limits_{N \in \mathbb{N}}|\sum\limits_{n=1}^{N}x_n|$. One can easily see that this is a norm on $\ell^1$. I was wondering if this space is now complete. ...
2
votes
3answers
48 views

Why would this series Absolutely Converge using Root Test?

So I was working on this problem and I got diverge, since my answer was greater than 1. The Limit was > 1, using the root test. $$\sum\limits_{n=4}^\infty (1 +\frac{1}{n})^{-n^2}$$ I ended up with ...
0
votes
1answer
30 views

Could I get an explanation on why this would conditionally converge?

$$\sum\limits_{n=2}^\infty \frac{\cos(n\pi)}{\ln(n)^2}$$ I'm not sure how this would conditionally converge, according to my calculations I would assume it's absolutely converge.
1
vote
1answer
58 views

$\sum u_v$ converges absolutely iff $\sum \log(1 + u_v)$ converges absolutely

I have trouble understanding the following proof of a fact in complex analysis. Assume $(u_v)_{v\geq1}$ is a sequence of complex numbers and $(u_v) \neq -1$ for all $v$. Then we have the following ...
0
votes
1answer
46 views

Switching Integral and Sum

I want to proof that I can switch this Sum and Integral $\sum\limits_{n=1}^\infty\int\limits_{0}^\infty t^{z-1} e^{-nt}dt~~$ for $~ 1 < Re(z) $ to sum it after over n. I tried to use the ...
0
votes
1answer
26 views

Shouldn't we check for conditionally convergent in ratio test done to see the intervals of convergence in power series? [closed]

(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute value ...
0
votes
1answer
42 views

Can the Ratio & Root Tests show divergence directly?

An infinite series ⅀ $a_n$ is absolutely convergent if ⅀ $|a_n|$ is convergent. However, just because the absolute value of a series isn't convergent by some test doesn't mean it can't be ...
0
votes
3answers
32 views

Is $\sum_{n=1}^{\infty} \frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$ divergent, conditionally convergent, or absolutely convergent?

The first thing I did was create a sequence ($A$) for what was inside of the sum, then I created another sequence ($B$) that is related to $A$. $$A=\frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$$ $$B=\frac{1}{n^2}...
1
vote
2answers
194 views

Why are the sums of absolutely convergent series not affected by changing the order of summation?

Here is my attempt to understand this: Let $$ \sum_{n=1}^{\infty}\left(-1\right)^{\left(n+1\right)}a_n $$ be an alternating series now the infinite sum is defined as the limit of the sequence of ...
0
votes
2answers
61 views

Proof for radius of convergence

I have the following two problems related to ordinary differential equations (power series) a. Define the radius of convergence of a series $\sum_{n=0}^{\infty}{a_{n}(x-x_{0})}^n$ for general $(...
1
vote
1answer
64 views

$\sum |a_n|<\infty$ and $|\sum b_n|<\infty$ implies $|\sum a_n b_n| <\infty$

Suppose $\sum_{n} a_n$ converges absolutely and $\sum_{n} b_n$ is any convergent series. Then $$\sum_n a_nb_n$$ is convergent. Proof: Since $\sum_{n} b_n$ is convergent, we can choose $N_0$ s.t $\...
0
votes
1answer
63 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
0
votes
0answers
22 views

Divergence of $\sum_{n=0}^\infty |\sin \omega n|$

I'm looking for a simple argument to show that $$ \sum_{n=0}^\infty |\sin \omega n| $$ does not converge for $\omega \neq k \pi$, $k \in \mathbb{Z}$. If $\omega = \frac{1}{b}\pi$ with $b \in \...
1
vote
1answer
49 views

Why does this inequality hold? (“Complex Analysis” by Kunihiko Kodaira.)

I am reading "Complex Analysis" by Kunihiko Kodaira. I cannot understand why the following inequality holds. Please tell me the reason why this inequality holds.
5
votes
1answer
62 views

If every rearrangement of the series converges uniformly then the series converges absolutely uniformly

Let $I \subset \mathbb{R}$ and for $\forall n \in \mathbb{N}: f_n \in C(I, \mathbb{R})$. Prove that if for any $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ bijection, the series $$\sum_n f_{\sigma(n)}$$ ...
0
votes
0answers
39 views

The proof of Taylor's theorem

Studying Analysis with 'Principles of Mathematical Analysis Third edition' written by Walter Rudin, I got some trouble from the proof of Taylor's theorem. The theorem and the proof from the book are ...
0
votes
1answer
36 views

Convergence when ratio test=1

When using the ratio test for absolute convergence of a series $\sum_{n=1}^\infty a_{n}$, if the limit of the ratio $$|a_{n+1}|/|a_{n}|=1$$ when $n \rightarrow \infty$, the fate of the series is ...
0
votes
2answers
37 views

Find the convergence radius for this power series

The Problem: Find the convergence radius of $\sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{(n)(n+1)}{2}}$ My attempts to find a solution I apply either the ratio test and end up with this ...
1
vote
1answer
40 views

Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
0
votes
1answer
31 views

Prove that $\sum_{i \in S} c_i \leq \sum_{i \in S'} c_i$ if $S \subset S' \subset \mathbb{N}$.

Is the following proof correct or not? Please tell me better proof. Let $\{c_n\}$ be a sequence of positive numbers such that $\sum c_n$ converges. Let $S \subset S' \subset \mathbb{N}$. ...
1
vote
0answers
45 views

About $f(x) = \sum_{x_n < x} c_n$ in Remark 4.31 on p.97 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. In Remark 4.31 on p.97, Rudin wrote this symbol $$\sum_{x_n < x} c_n.$$ What is the definition of this symbol ...
0
votes
3answers
59 views

Concluding whether $(y_n)_n$ is a bounded sequence

Suppose $(y_n)_n$ is a sequence in $\mathbb{C}$ with the following property: for each sequence $(x_n)_n$ in $\mathbb{C}$ for which the series $\sum_n x_n$ converges absolutely, also the series $\sum_n ...
0
votes
0answers
33 views

Convergence of $a_k=\sum_{k=5}^{+\infty}(-1)^k({3\over2})^{-k}(k^2+5)\sin(k+5)$

Firstly, $\lim_{k \to +\infty}a_k=0$,so necessary condition for convergence is satisfied.If we start to study absolute convergence we have : $$a_k=({3\over2})^{-k}(k^2+5)|\sin(k+5)|\leq({3\over2})^{-k}...
-1
votes
1answer
65 views

If $a_n$ doesn't have any subsequence that converges, can $|a_n|$ converge?

If a sequence $a_n$, $n\in\mathbb{N}$ doesn't have any convergent subsequence can $|a_n|\rightarrow a$, $a\in[0,\infty)$? My intuition says that this isn't possible but I'm not sure how to prove it..
1
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0answers
18 views

absolut and uniform convergence of complex power series on an open disk imply absolut and local uniform convergence on the whole disk of convergence [duplicate]

I'm stuck in the proof of the following theorem of complex analysis: There is a $R\in\mathbb{R}_{0}^{+}\cup\{\infty\}$ s.t.: (i) $\sum_{k=0}^{\infty} a_k (z-z_0)^k$ converges absolutely and ...
1
vote
0answers
27 views

On the convergence of a two-sided series

Background Let $\left\{ {{a_n}} \right\}_{ - \infty }^\infty $ be a two sided sequence (is there a more proper term?) of complex numbers. As far as I know (please correct me if I am wrong) we say ...
-1
votes
2answers
55 views

Find radius of convergence of the power series.

Find the radius of convergence of power series $$ \sum_{n=0}^{\infty} 2^{2n} x^{n^2}$$ A)1 B)2 C) 4 D)1/4 I try to apply ratio and root test ( Cauchy–Hadamard theorem ) .but they ...
2
votes
2answers
56 views

Let $X$ be a Banach space then every absolutely convergent series in $X$ converges in $X$

my trial Let $\sum x_k$ be absolutely convergent in $X$ $\implies$ $\sum \|x_k \|$ converges in $\mathbb{R}$ $\implies$ $\forall \epsilon >0, \exists N(\epsilon)$ st $\forall n>N(\epsilon)$. ...
2
votes
2answers
56 views

Hint requested for: If $\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $a<|x_0|$?

I would like to prove If $\displaystyle\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $0<a<|x_0|$. (Sorry not enough ...
0
votes
1answer
64 views

Difficulty regarding understanding a proof of multiplication of infinite series

I am referring to article no $3.50$, from Principles of Mathematical Analysis by Walter Rudin. The theorem is Let $\sum_{n=0}^\infty a_n=A$ and $\sum_{n=0}^\infty b_n=B$ be two convergent series. ...
1
vote
1answer
30 views

Convergence of sum using ratio test

Determine the values of x ∈ R for which the following series converge: $\sum_1^\infty \frac{x^nn^n}{n!}$ My attempt: I used the ratio test to obtain $|x|< \frac{1}{e}$ as the interval where it ...
0
votes
1answer
32 views

Dirichlet test for complex sequences

The sum $$\sum_{k=1}^\infty a_k b_k$$ converges when $a_k$ is monotonically decreasing and $$B_n=\sum_{k=1}^n b_k$$ is finite/bounded $\forall n$. This follows from summation by parts. I'm now ...
12
votes
1answer
348 views

Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in ...
0
votes
3answers
46 views

Show with the direct Comparison Test that $\sqrt[n]{|a_n|}\leq\theta$ converges absolutely

Let $\sum_{n=1}^{n=\infty}{a_n}$ be an infinite series of real numbers. There is a $\theta$ with $0<\theta<1$ and a $n_0 > 0$, so $$\sqrt[n]{|a_n|}\leq\theta$$ for all $n \ge n_0$. Show that ...
0
votes
1answer
46 views

Absolute or conditional convergence?

Determine whether the series: $$\sum_{n=1}^\infty (-1)^n \frac {2n^2+3n+4} {2n^4 + 3}$$ converges absolutely, conditionally or diverges. I know the series converges conditionally using alternating ...
1
vote
5answers
131 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$.

I am analizing the convergence, absolute convergence and conditional convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$. I proved already that the series $\sum_{n=1}^{\...
0
votes
2answers
57 views

Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test

Image of My Work I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks P.S. if my ...
0
votes
0answers
46 views

Does convergence of $\Gamma(x)$ imply convergence of $\Gamma(z)$? Is it generalisable?

If we have the gamma function in integral form $\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$ and have proven that it converges for real $x>0$ (to a point), can we then immediately conclude that $...