Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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 ...
-1
votes
0answers
21 views

A problem on the basics of convergence of a series

I was taught that if a series converge , for an example if {an} converges then lim ∑an = 0 But again I was taught that , an = (-1)^n is absolutely convergent and it converges to 1. I think it ...
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
votes
0answers
14 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
votes
0answers
31 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
43 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
21 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
36 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
29 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
54 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
43 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
22 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
41 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
180 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
56 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
53 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
60 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
20 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
45 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
35 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
35 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
36 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
38 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
63 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
vote
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
24 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
50 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
53 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
61 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
29 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
334 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
40 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
129 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
52 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 $...
0
votes
0answers
29 views

$\sqrt{n}(x_n-x_0) \to N(0,\sigma_0^2)$ in distribution implies $x_n \to x_0$ almost surely

Is this true? I figured that if not, there will some positive probability $\sigma$ that $\sqrt{n}(x_n-x_0)$ takes $\sqrt{M} \cdot \epsilon$ for infinitely many large $M$. Even though this "blowing up" ...
1
vote
2answers
36 views

Find the range of convergence of the series$\,\,\sum_{n=0}^\infty {\frac{z^n}{1+z^{2n}}}$

The series I have is $$\displaystyle\sum_{n=0}^\infty {\dfrac{z^n}{1+z^{2n}}}$$ The same series with absolute values is: $$\displaystyle\sum_{n=0}^\infty {\dfrac{|z|^n}{1+|z|^{2n}}}$$ Using D'...
7
votes
2answers
317 views

Using the root test when the limit does not exist

I used the root test for the series $$ \sum_{n=1}^{\infty} \left(\frac{\cos n}{2}\right)^n. $$ I showed that $$ 0 \le \left|\frac{\cos(n)}{2}\right| \le \frac{1}{2} \implies \lim_{n\to\infty}\left|\...
0
votes
1answer
28 views

Absolutely convergent series of complex functions.

I have to do the following excercise: Let $\{f_n(z)\}_{n\in\mathbb{N}}$ a sequence of complex functions, and let $\sum_{n=1}^\infty f_n(z)$. Prove that: if $\sum_{n=1}^\infty |f_n(z)|$ converges, ...
1
vote
1answer
47 views

If $\sum g_n(x)$ converges uniformly and absolutely and $|f_n(x)|\leq |g_n(x)|$ show that $\sum f_n(x)$ converges uniformly and absolutely.

I do not know how to prove if the statement above is true. I know i can use the Cauchy criterion i.e. $|\sum_{n\rightarrow m}f_n(x)|\leq\sum_{n\rightarrow m}|f_n(x)|\leq \sum_{n\rightarrow m}|g_n(x)...