Questions tagged [absolute-convergence]

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

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$\sum_{n=1}^\infty \int_{\mathbb R}|f_n| d\lambda<\infty \Rightarrow \sum_{n=1}^\infty f_n$ converges almost everywhere absolutely [closed]

Let $(f_n)_{n\in\mathbb N}$ be a sequence of measurable functions on $\mathbb R$ with $\sum_{n=1}^\infty \int_{\mathbb R}|f_n|d\lambda <\infty$. Show that $\sum_{n=1}^\infty f_n$ converges almost ...
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  • 47
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2 answers
42 views

Radius of convergence of power series when $x+y \leq n$

I can't figure out how to solve this problem... Any help would be thoroughly appreciated. Let $a_n$ be the number of pairs of $(x, y)$ such that $x + y \leq n$, where $x$ and $y$ are non-negative ...
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  • 25
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22 views

absolutely convergent integral and an additional question on asymptotics

I want to show that the integral $$\int_1^{\infty}e^{-a(x-\ln(x))}\frac{x}{e^{cx}(1+x^2)}dx$$ is absolutely convergent by the dominated convergence theorem. So I figured that since $$e^{-a(x-\ln(x))}\...
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  • 31
0 votes
1 answer
40 views

Absolutely convergence of sum of a sequence of complex numbers over an arbitrary set

I read on notes of a course on differential geometry the following statement but it doesn't seem so clear to me. Let $I$ be a set and $b: I \rightarrow \mathbb{C}$ a map, where $b_i=b(i)$, namely we ...
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2 votes
0 answers
49 views

Prove/disprove a claim in calculus

Prove/disprove: If $\sum\limits_{n=0}^{\infty} a_n$ converges then $\sum\limits_{n=0}^{\infty} a_n^{2022}$ absolutely converges (i.e. $\sum |a_n^{2022}|$ converges). If I take $a_n=\dfrac{(-1)^{n+1}...
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  • 249
1 vote
2 answers
40 views

Find it's radius of convergence, interval of convergence of the power series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+2)^n}{n2^n}$

Consider the power series $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+2)^n}{n2^n}$$ Find it's radius of convergence, interval of convergence. Also find for which $x$ the series is absolutely convergent ...
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  • 1,446
9 votes
3 answers
176 views

Is there a sequence so that $\sum |a_n|=\infty$ and $\sum a_n \cos(nx)$ and $\sum a_n \sin(nx)$ converge everywhere?

Let the series be $(s_n a_n)_n$ where $s_n\in\{-1,1\}$ and $a_n$ decreases to $0$ instead for convenience. If $s_n$ is eventually periodic you can choose an $x$ which is a rational multiple of $\pi$ ...
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-1 votes
1 answer
49 views

How can I calculate at which distance a moving object stops in this physics emulation? [closed]

There is an object which has an initial velocity of $`V_1`$ and has an acceleration of $`V^2 k+d+b`$ in the opposite direction, where k,d,b are constants. What will ...
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1 vote
1 answer
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Let ${a_n}_{n \in \mathbb Z_+}$ be a sequence of complex numbers. Suppose $Z_+ = \cup_1^{\infty} B_k$...

Let ${a_n}_{n \in \mathbb Z_+}$ be a sequence of complex numbers. Suppose $\mathbb Z_+ = \cup_1^{\infty} B_k$ where each $B_k$ is infinite and $B_k \cap B_{k'} = \emptyset$ if $k \ne k'$. Prove that ...
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  • 103
2 votes
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A power series that converges conditionally for all points on the radius of convergence?

For $a_n\in\Bbb C$ let $$f(z) = \sum_{k=0}^\infty a_n z^n \tag 1$$ be a power series with radius of convergence of 1, and $a_n$ such that the series converges for all $z\in\Bbb C$ with $|z|=1$. What's ...
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1 vote
1 answer
65 views

Absolutely convergent series implies being an analytic function

Consider the Riemann zeta function $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}},$$ This series converges absolutely on $\text{Re}(s)>1$. I have seen in multiple literature that this implies that ...
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  • 65
3 votes
3 answers
202 views

Does $ \sum_{k=1}^{n} \frac{(n-k)^k}{k!} $ have a closed-form expression in terms of $n \in \mathbb{N}$?

Does $ \sum_{k=1}^{n} \frac{(n-k)^k}{k!} $ have a closed-form expression in terms of $n \in \mathbb{N}$? It seems to grow a bit faster than $e^{0.5n}$, but there's clearly more to it, and I don't know ...
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1 vote
0 answers
13 views

absolut convergence of the Laplace transform of a intensity measure

Let $\xi$ be a point process and $\mu$ its intensity measure, i.e. $\mu(\cdot)=\mathbb{E}[\xi(\cdot)]$. The Laplace transform of $\mu$: $\mathcal{L}\mu(z)=\int_{0}^{\infty}e^{-zx}\mu(dx)$ converges ...
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  • 27
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0 answers
32 views

A double series nature

here is defined a double series: $\displaystyle \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m} \frac{\cos(b\ln(2i))\cos(b\ln(2j))}{(2i)^{a}(2j)^{a}}-\frac{\cos(b\ln(2i-1))\cos(b\ln(2j-1))}{(2i-1)^{a}(2j-1)...
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  • 1
1 vote
1 answer
36 views

Confusion about convergence for the logarithm of a matrix

How can I prove that for a matrix $A$, $$ \text{log}A = \sum_{m=1}^\infty (-1)^{m+1} \frac{(A-I)^m}{m} $$ is absolutely convergent if $||A - I|| < 1$? (I'm using the Hilbert-Schmidt norm.) In Hall'...
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  • 197
0 votes
1 answer
27 views

Determine whether the following infinite series is absolutely convergent, conditionally convergent, or divergent?

$\sum_{n=0}^\infty(-1)^n(\sqrt{n^5}-\sqrt{n^5-n^2})$ My attempt: I tried to use Leibniz's test and found that a_n approaches 0 when sent to infinity, but I cannot tell if it monotonic decreases as my ...
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0 votes
0 answers
11 views

Clarification in a question regarding vector space and series convergence

The question I am referring to is Hassani's Mathematical Physics Problem 2.18: Using the Schwarz Inequality to show that if $\{\alpha_i\}_{i=1}^{\infty}$ and $\{\beta_i\}_{i=1}^{\infty}$ is in $\...
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2 votes
1 answer
99 views

Natural boundary for $\sum_{n=0}^{\infty}\frac{z^{n!}}{n!}$

In my complex variable course we are studying series convergence, using comparison test I determined that the series \begin{equation} \sum_{n=0}^{\infty}\frac{z^{n!}}{n!} \end{equation} has a ...
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0 votes
1 answer
31 views

MGF exists and yet derivative does not

Consider the Moment Generating Function of the Uniform distribution $U(a,b)$ given by $$M(t)=\frac{e^{tb}-e^{ta}}{t(b-a)} \mbox{if }t\ne 0; M(0)=1.$$ Now, it is well known that if MGF is finite in an ...
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  • 9,857
1 vote
2 answers
68 views

Ratio test (missing proof step): $\lim_\limits{n\rightarrow \infty}\left| \frac{a_{n+1}}{a_n} \right|<1 \Rightarrow \sum_\limits{k=0}^n a_k$ converges

I'm trying to proof the ratio test for convergence of sum of a sequence: $l :=\lim_\limits{n\rightarrow \infty}\left| \frac{a_{n+1}}{a_n} \right| <1 \Rightarrow s_n := \sum_\limits{k=0}^n a_k$ ...
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2 votes
1 answer
60 views

Existence of MGF in an interval containing $0$ implies that moment of all orders exist

This is not a duplicate of this (although it refers to the same problem), and maybe a very basic question so kindly bear with me. Suppose the Moment Generating Function (MGF) $M(s)$ is finite for some ...
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  • 9,857
0 votes
1 answer
52 views

Given $\sum^{∞}_{n=1} \; \frac{(-1)^{n}}{\ln(n)^{2}}$, how to determine if it is convergent [closed]

I have tried the integral test, and I don't know what to do with the numerator that is not defined when n approaches infinity. The thing is that I know that I could try to "take"/find out if ...
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  • 13
0 votes
1 answer
28 views

Using sum of absolute values as dominating function to justify dominated convergence theorem and interchange of summation/integration

I'm trying to show that: $\int_{0}^{\infty} e^{-u} \, \sum_{n=0}^{\infty} \dfrac{u^n}{2^{n} \, (n!)^2} \; du \;= \; \sum_{n=0}^{\infty} \dfrac{1}{2^{n}\, (n!)^2} \; \int_{0}^{\infty} e^{-u} \, u^n \, ...
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0 votes
0 answers
35 views

Interchange order of summation and integration using dominated value theorem and absolute convergence

I'm trying to show that: $\int_{0}^{\infty} e^{-u} \, \sum_{n=0}^{\infty} \dfrac{u^n}{2^{n} \, (n!)^2} \; du \;= \; \sum_{n=0}^{\infty} \dfrac{1}{2^{n}\, (n!)^2} \; \int_{0}^{\infty} e^{-u} \, u^n \, ...
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1 vote
0 answers
20 views

Showing this sum is absolutely convergent using the ratio test.

I'm trying to show that $ \sum_{n=0}^{\infty} \dfrac{(-1)^n \, x^{2n+1}}{2^n\,n!}$ is absolutely convergent for all $x$. Using the ratio test I get: $\lim_{n \to \infty} \left|\dfrac{f_{n+1}(x)}{f_n(x)...
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0 votes
1 answer
34 views

An attempt proof of Dirichlet's test

I'm trying to learn the convergence of $\sum_{n\ge 1} z^n/n$ when $|z|=1$ but $z\not =1$. This site points out the existence of Dirichlet's test when discussing towards this problem. The test states ...
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20 votes
5 answers
887 views

An absolutely convergent series of rational numbers which does not converge to a rational number [duplicate]

A standard theorem concerning series of real numbers states that every absolutely convergent series of real numbers converges. I would like to know a counterexample to this statement when we are ...
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1 vote
1 answer
45 views

Transforming bounded sequences with the same limits

Let $(a_{n,k}: n,k \ge 1)$ and $(b_{n,k}: n,k\ge 1)$ be nonnegative reals with the property that: $\lim_{n\to \infty}\sum_{k=1}^\infty a_{n,k}=\lim_{n\to \infty}\sum_{k=1}^\infty b_{n,k}=1$; for ...
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1 vote
0 answers
40 views

Show that the sum of the random variables converges absolutely almost surely for given conditions.

Here is my question and here is my partial solution. I am uncertain if I proved the hint correctly, and I am basically stuck now after proving the hint. I have seen a solution to a similar question ...
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1 vote
1 answer
57 views

Convergence on sum of sequence

Let $\{ a_n \}^\infty_1 \subseteq \Bbb{R}$ be a sequence. If for any recursively enumerable set $S = \{k_1, k_2, ...\} \subseteq \Bbb{N}$, where $k_1 < k_2 < ...$ is an infinite ascending chain, ...
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  • 1,497
0 votes
0 answers
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I cannot understand why we need to prove this theorem only for a non-negative series $\sum a_n$ ("A First Course in Analysis vol.1" by Sin Hitotumatu)

I am reading "A First Course in Analysis vol.1" (in Japanese) by Sin Hitotumatu. The following theorem is in this book. Theorem 7.6: Let $\mathbb{N}=N_1\cup N_2\cup\cdots$ and $N_i\cap N_j=\...
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  • 6,508
6 votes
0 answers
168 views

Does $\ \sum_{n=1}^{\infty} \frac{\sin(2^n x)}{n}\ $ converge for all $x\ ?$

Does $\ \displaystyle\sum_{n=1}^{\infty} \frac{\sin(2^n x)}{n}\ $ converge for all $x\ ?$ It obviously converges for any $x\ $ of the form $\ 2^mk \pi\ $ where $\ m,k\in\mathbb{Z},\ $ but for any ...
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0 votes
0 answers
19 views

Infinite product over primes' convergence with rearrangement

If we know that $W_m(k)$ is multiplicative over $k$ (if $\gcd(k_1,k_2)=1$ then $W_m(k_1k_2)=W_m(k_1)W_m(k_2)$), and $W_m(k)=0$ if $p^2|k$, and that $|W_m(k)|\le c\cdot \frac{m}{k^{3/2}}$. So that the ...
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-3 votes
1 answer
118 views

1-1+1-1+... = k+1/2?

Let $$F(s) = \sum_{n=0}^{\infty} f_{n}(s)$$ be a complex analytical function defined by a series (not necessarily a power series) that absolutely converges on an open set $s \in U$. Assume that the ...
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  • 553
2 votes
1 answer
210 views

Integral representation of Bessel function $J_1(x)$

In "The Handbook of Mathematical Functions" by Abramovitz and Stegun, according to Eq. 9.1.24, \begin{align} J_0(x)=&\frac{2}{\pi}\int_{1}^\infty \frac{\sin(xt)}{\sqrt{t^2-1}}dt,\quad x&...
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0 votes
1 answer
44 views

How to study if an alternating series with squareroot n minus n is absolute convergent or simply convergent?

So, I am supposed to tell if the following series is absolute convergent or simply convergent: $$\sum(-1)^n\left(\sqrt{n^2+1}-n\right)$$ To study ther absolute convergence I know that I need to study ...
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0 votes
0 answers
35 views

Fourier series converges absolutely almost everywhere to smooth function. Maybe also everywhere?

Setting: Let $A : \mathbb R^2 \to \mathbb R$ be a function with the property that for fixed $y \in \mathbb R$ the restriction $x\mapsto A(x,y)$ is smooth on $\mathbb R$. Further, we have for every $m \...
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9 votes
0 answers
134 views

Group of all permutations of $\mathbb N$ that don't change the limit of series

Clearly all bijections functions $\varphi : \mathbb N \to \mathbb N$ form a group, let's call it $S(\mathbb N)$. Now let's define $H \subset S(\mathbb N)$ to be the set of all elements $\varphi \in S(\...
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4 votes
1 answer
104 views

Does uniform convergence imply absolute convergence for Fourier series?

Let $f : \mathbb R \to \mathbb R$ be a continuous $1$-periodic function. We can compute the Fourier coefficients $$a_n := \int_0^1 f(x)\exp(-2\pi i nx) dx.$$ If $$\sum_{n \in \mathbb Z} |a_n| < \...
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0 votes
1 answer
39 views

Which conditions makes integrals of the form $\int_{-\infty}^\infty \left|\frac{\mathbb{P}(w^a)}{\mathbb{P}(w^b)} \right|dw < \infty$ converge?

Which conditions must fulfill an integral to be convergent $\int_{-\infty}^\infty \left|\frac{\mathbb{P}(w^a)}{\mathbb{P}(w^b)} \right|dw < \infty$ with $\mathbb{P}(w^m)$ a polynomial with ...
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  • 1,031
1 vote
1 answer
60 views

Does $\sum^{∞}_{k=1} \frac{4k^k}{2^{k^2}}$ converge absolutely?

I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally. $$\sum^{∞}_{k=1} \frac{4k^k}{2^{k^2}}$$ I decided to ...
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3 votes
1 answer
81 views

Does series converge absolutely or conditionally?

I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally. $$\sum^{∞}_{k=1} \frac{\sin(2k^2+1)}{k^{3/2}}$$ I ...
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1 vote
3 answers
290 views

Example of an absolutely summable series that is not summable

When I encountered Banach spaces I was presented with some proofs that link completeness, vector spaces, series and sequences (of partial sums). In particular I was presented with the following ...
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0 votes
1 answer
123 views

Why Wolfram-Alpha (W-A) said that $\lim_{x \to \infty} e^{\sqrt{x}}\cdot\text{sinc}(x) = 0\,$?? [closed]

Why Wolfram-Alpha (W-A) said that $\lim_{x \to \infty} e^{\sqrt{x}}\cdot\text{sinc}(x) = 0\,$, if $\,\lim_{x \to \infty} \frac{e^{\sqrt{x}}}{x^n} = \infty$ for any $n\gg 0$?? I have tried the first ...
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  • 1,031
2 votes
1 answer
89 views

Determine if the series converges absolutely, converges conditionally, or diverges.

Determine if the series converges absolutely, converges conditionally, or diverges. Find the exact value for the sum of the convergent series. $$ 1-\frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5}...
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0 votes
1 answer
343 views

Determine if the series converges absolutely, conditionally, or diverges.

Determine if the series converges absolutely, converges conditionally, or diverges. Find the exact value for the sum of the convergent series. $$1-\frac{1}{5} - \frac{1}{5^2} + \frac{1}{5^3} - \frac{1}...
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0 votes
0 answers
49 views

Kunita-Watanabe decomposition

proof Hello, I have a problem with this part of the proof to the Kunita-Watanabe decomposition. I don't understand how they use Lemma 1.69 to conclude that $\mathcal{G}$ is closed in the Hilbert space ...
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3 votes
2 answers
68 views

Convergence and absolute convergence of alternating series $\sum\int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t}dt$

I have to determine the convergence (later absolute convergence) of the alternating series with the general term $$v_n=\int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t}dt$$ using the usual alternating series ...
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0 votes
1 answer
47 views

Uniform convergence of series with nonnegative terms implies uniform convergence of terms to zero [closed]

If I tell you that $S_n(x) = \sum^n_{k=0} \lvert f_k(x)\rvert $ converges uniformly, does it follow that $\lvert f_k(x)\rvert$ converge uniformly to zero? Assume that $f_k$ are real-valued functions ...
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0 votes
0 answers
42 views

Explanation of the proof of absolute convergence of $ \zeta(s, a) $

Here is Theorem 12.1 of Introduction to Analytic Number Theory by Apostol - Theorem 12.1 The series for $ \zeta(s, a) $ converges absolutely for $ \sigma > 1 $. The convergence is uniform in every ...
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