Questions tagged [absolute-convergence]
This tag is for questions related to absolute convergence of a series.
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Find all the values $x\in\mathbb{R}$ where the Series converge
$$
\sum_{n=1}^{\infty}\frac{\sqrt{n}+3n}{2^{n}+5n}(x-1)^{n}
$$
I calculate the limit $n\to\infty$ with the D'Alambert ratio test, and the series converges in the interval $-1<x<3$:
$$
\sum_{n=1}^...
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Existence of absolutely convergent subseries given the base sequence converges to zero
My Real Analysis final is coming up and I'd like to practice working with sequences and series, so I picked a practice problem and tried working it out. The statement is the following:
Let $ (x_n)_n $...
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Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$
Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
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Bounding of this series [duplicate]
Prove this series converges absolutely.
$$\sum_{k = 1}^{+\infty} \frac{(x^2-7x+6)^n}{n^2 6^{n+2}}$$
Attempts
As in the comments, it doesn't say where. But I believe it's meant for $x \in (0, 3) \cup (...
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How To Prove That The Absolute Convergence Of The One-Step Transition Matrix?
I am trying to prove that the n-step transition matrix in a Markov Chain satisfies the condition that the sum of each row is one (ie, each row is a valid probability distribution). I have done this ...
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Proof of a Limit related to Gauss' Convergence test
So this is the question:
if the series $\sum_{n=1}^{\infty} a_n$ is such that $$\frac{a_n}{a_{n+1}} = 1 + \frac pn + \alpha_n$$ and the series $\sum_{n=1}^{\infty} \alpha_n$ converges absolutely, ...
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Absolute convergence of series $\sum_{n=1}^{\infty }(\frac{\cos(n)}{\ln(n^{2n}+n^2)}+1-\cos(\frac{1}{n}))$
I have shown the convergence of the series by using Dirichlet's test to show that the first summand $\frac{\cos(n)}{\ln(n^{2n}+n^2)}$ converge and the comparison test to show that $1-\cos(\frac{1}{n})$...
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Weierstrass' M-test and convergence of series
I came across with the following proof about the convergence of Dirichlet L-series but I have troubles understanding it:
Let $\delta >0$ and $f:\mathbb{Z}\rightarrow \mathbb{C}$ satisfy $|f(n)|\leq ...
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Convergence in $𝐿^1$ and continuos CDF , implies converges in $𝐿^1$ for the corresponding binarized succession?
Let $X_n$ a succession of random variables with $\{x \in \mathbb{R}: \mathbb{P}(X_n=x)>0\} \subseteq[0,1]$ $\forall n$, and X a random variable with $\{x \in \mathbb{R}: \mathbb{P}(X=x)>0\} \...
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I want to know various ways to check convergence of infinite product
In complex analysis class I learned some condition that makes infinite product converge.
Theorem : If $\sum |a_n|$ converges, then the infinite product $\prod(1+a_n)$ also converges.
The proof is ...
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Associativity of infinite products
It is well-known that if $\sum_{n=1}^\infty a_n$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\...
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Absolute Convergence of Fourier Series Proof
I am looking at the following theorem
Let $f$ and $g$ be two piecewise continuous, periodic functions with the same period $p$ and with Fourier series
$$\begin{align}
\mathcal{F}[f]|_{t} = \sum_{k=-\...
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Absolute and conditional convergence of a non-alternating series
I tried all tests such as d'Alembert test, Cauchy test, Leibniz test,... but i could't determine convergence of this series:
$$\sum_{n=2}^{\infty }\frac{(-1)^n}{\sqrt{n}+(-1)^n}$$
Can you help me!!!
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Convergence of series from inverse of Cauchy product
The Cauchy product of two real or complex infinite series $\sum_{n\in\mathbb{N}} a_n$ and $\sum_{n\in\mathbb{N}} b_n$ is defined as:
$$ \forall n\in\mathbb{N}, c_n = \sum_{k=0}^n a_k b_{n-k} $$
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How to Prove the Divergence of an Improper Integral Involving Absolute Value
I'm working on understanding the convergence properties of certain improper integrals and encountered the following integral:
$$\int_{0}^{\infty} \left| \frac{\cos(x)}{\sqrt{x}} \right| \, dx$$
I ...
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