Questions tagged [absolute-convergence]
This tag is for questions related to absolute convergence of a series.
742 questions
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Is "the nth logarithm test" for series' convergence reliable?
Is it possible to determine the convergence of a series $\sum a_n$ by evaluating $$\lim_{n \to \infty} \log_n(a_n)$$ and comparing the result to $-1$? Specifically, if this limit is less than $-1$, ...
- 58
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2
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48
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Unconditional convergence and analytic map
Here is the definition of an analytic map.
ANALYTIC MAP
Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if
$$
f(x)=\sum_{k=0}^{\infty}a_k(x) \tag 1
$$
where
for ...
2
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2
answers
153
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Continuity of $\sum_{n = 1}^{\infty} e^{-nx}\sin{nx}$.
I was doing a problem where I was asked to show the continuity of
$$\sum_{n = 1}^{\infty} e^{-nx}\sin{nx},$$
for $x>0$.
My approach was to consider the sequence $(\sigma_{n}(x))$ of partial sums of ...
- 342
0
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1
answer
125
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Split a series in a sum of series
Suppose that $\sum_{n = 0}^\infty a_n$ is absolutely convergent and $\{P_1,\ldots, P_r\}$ is a partition of $\mathbb{N}$ (i.e. $\bigcup_{i=1}^r P_i = \mathbb{N}$ and $P_i \cap P_j = \emptyset$ for any ...
- 1,482
6
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3
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384
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Probability with a Converging Infinite Sum
At a fairground, one big prize is being given away by a random chance game, such as spinning a wheel. A finite number of players line up in a queue and take turns with the game. If the person at the ...
5
votes
1
answer
150
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Absolute convergence of a series in a Banach space.
Consider this classic proposition for nets from Banach Algebra Techniques in Operator Theory by Douglas:
1.8 Definition Let $\{f_{\alpha}\}_{\alpha \in A}$ be a set of vectors in the Banach space $\...
- 2,757
3
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1
answer
72
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Absolute/conditional convergence of $ \sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr) $
Examine the absolute/conditional convergence of the series
$$
\sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr)
$$
I can't seem to bound it from above, so I'm ...
- 437
3
votes
1
answer
111
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From formal power series to analytical ones
What is the best way to describe the relation between the ring of formal power series $\mathbb C[[X]]$ and the actual complex power series?
I understand both separately, but I have never seen them ...
- 1,522
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0
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124
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Showing the decimal expansion of $\sqrt 2$ is a Cauchy sequence
I've recently been teaching myself about Cauchy sequences and I'm trying to understand a certain proof that the decimal expansion of $\sqrt{2}$ is a Cauchy sequence.
This proof uses the following ...
- 561
2
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1
answer
103
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Two definitions of summable series in a Hilbert space
I am trying to prove a comment made on page 18 in Halmos's Introduction to Hilbert Spaces and The Theory of Spectral Multiplicity. For below, assume $X$ is an arbitrary Hilbert space. I am rephrasing ...
- 1,070
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0
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60
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Converging absolutely and uniformly versus converging uniformly absolutely [duplicate]
I'm writing because I suspect Conway's "Functions of one complex variable" makes a minor mistake with its terminology, but I'm tired and unsure if I'm missing something.
In the chapter on ...
- 48.2k
3
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0
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224
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Please suggest a book that explains this theorem. We can use this theorem when we prove $e^{iz}=\cos z+i \sin z$. (Sin Hitotumatu's analysis book.)
I am reading "Introduction to Analysis 1" (in Japanese) by Sin Hitotumatu.
This book contains the following theorem.
I found this theorem interesting.
For example we can use this theorem ...
- 10.4k
0
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1
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101
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Why is absolute convergence necessary for the Rearrangement Theorem?
The following proof of the rearrangement theorem is from Bartle and Sherbert's Introduction to Real Analysis, 3rd edition. I do not see where absolute convergence was used in the proof.
9.1.5 ...
- 25
3
votes
1
answer
97
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Absolute convergence of $\mathop\sum\limits_{{n = 1}}^{\infty }{a}_{n}$
For $\mathop\sum\limits_{{n = 1}}^{\infty }{a}_{n}$, if for any subsequence $\left\{ {a}_{{n}_{k}}\right\}$, we have the convergences of $\mathop\sum\limits_{{k = 1}}^{\infty }{a}_{{n}_{k}}$ then by ...
- 327
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136
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Absolute convergence of a series of functions
I came across the definition of absolute convergence of a series of functions and I am unsure if I did understand said definition correctly.
Thus far, the only notion of absolute convergence I knew ...
- 359
1
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0
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44
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Double summation limit with denominator term
Is there any general theory talking about if the following limit
$$
\lim_{n \rightarrow \infty} 1/n \sum_{i=1}^{n} \frac{\sum_{j=1}^n f(x_i,x_j)}{\sum_{j=1}^n g(x_i,x_j)}
$$
converges to
$$
\int\frac{\...
- 159
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0
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66
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Show convergence of integral $\int\limits_1^\infty\frac{1}{P(x)}dx$
Let be $P(x)$ a polynomial of degree $n$,i.e. $P(x):=a_0+a_1x+\dotsc+a_nx^n$. We assume that all zero spots of $P(x)$ are negative. Show that $\int\limits_1^\infty\frac{1}{P(x)}dx$ converges ...
- 5,010
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46
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Proving the existence of the limit of the sequence given that the distance between the terms gets smaller
I have to prove that $\lim_{n\to\infty}x_n$ exists given $|x_{n+1}-x_n| = 2^{-n}$
I have proven this using Cauchy criterion: $\forall\epsilon>0 \space \exists N\in\mathbb{N} \space \forall m>n&...
- 327
1
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0
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36
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Express $a(s)$ in terms of $F(z)=\int_0^{\infty}a(s)z^sds$?
Let's consider a function $$F:z\mapsto F(z)=\int_0^{\infty}a(s)z^sds,$$ where $z$ is a real number in some small interval, say $(0,z_0)$. Also, the coefficients $a(s)$ are nonnegative, and they enjoy ...
- 351
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53
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If a series of holomorphic functions converges absolutely uniformly on a disk then the series of derivatives converges absolutely uniformly
Let $f_{i}$ be a collection of homolomorphic functions on a disk $\Delta$ such that $\sum_{i=1}^{\infty} |f_{i}(z)|$ converges uniformly on $\Delta$. Prove that $\sum_{i=0}^{\infty}|f'(z)|$ converges ...
- 153
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2
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71
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Find all the values $x\in\mathbb{R}$ where the Series converge
$$
\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}^...
- 51
2
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0
answers
44
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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 $...
- 556
-2
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1
answer
173
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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^{\...
- 23
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1
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56
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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 (...
- 3,675
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0
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31
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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 ...
- 185
3
votes
1
answer
122
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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, ...
- 695
1
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2
answers
147
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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 $1-\cos(\frac{1}{n})$...
- 45
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2
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457
views
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 $\...
- 1,700
1
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0
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58
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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
$$\begin{align}
\mathcal{F}[f]|_{t} = \sum_{k=-\...
0
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2
answers
62
views
Absolute and conditional convergence of a non-alternating series
I tried all tests such as d'Alembert test, Cauchy test, Leibniz test,... but i could't determine convergence of this series:
$$\sum_{n=2}^{\infty }\frac{(-1)^n}{\sqrt{n}+(-1)^n}$$
Can you help me!!!
4
votes
1
answer
193
views
Convergence of series from inverse of Cauchy product
The Cauchy product of two real or complex infinite series $\sum_{n\in\mathbb{N}} a_n$ and $\sum_{n\in\mathbb{N}} b_n$ is defined as:
$$ \forall n\in\mathbb{N}, c_n = \sum_{k=0}^n a_k b_{n-k} $$
...
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1
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1
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102
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How to Prove the Divergence of an Improper Integral Involving Absolute Value
I'm working on understanding the convergence properties of certain improper integrals and encountered the following integral:
$$\int_{0}^{\infty} \left| \frac{\cos(x)}{\sqrt{x}} \right| \, dx$$
I ...
- 11
2
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0
answers
76
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Rearranging conditionally convergent series without changing the limit
Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that the series $\sum a_n$ is conditionally convergent, i.e. the limit $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
- 1,057
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0
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54
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Convergence of $\sum |a_n|$
I have question about convergence of $\sum |a_n|$,
$1.$ If $\sum |a_n|$ converges then $\sum a_n^k$ converges of any $K \in N$,
my reasoning was, an $\sum |a_n| $ converges which means $\exists N$ ...
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1
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The implications of an absolutely convergent Infinite Series [closed]
Does the convergence of $|a_n|$ imply the convergence of $\sqrt[4]{\frac{|a_n|}{n^4}}$ ?
E.g. in R: $$\sum_{n=0}^{\infty} |a_n|\rightarrow a \in R \Rightarrow \sum_{n=0}^{\infty} \sqrt[4]{|\frac{...
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0
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For which $\{a_k\}_{k=1}^\infty$ does $\sum_{k=1}^\infty \frac{1}{a_k} f(x+a_k)$ converge absolutely for almost every $x\in \Bbb R$?
Question: Let $f\in L^1(\Bbb R)$. For which increasing sequences $\{a_k\}_{k=1}^\infty$ of positive real numbers does $$\sum_{k=1}^\infty \frac{1}{a_k} f(x+a_k)$$ converge absolutely for almost every $...
- 15k
1
vote
0
answers
51
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Find the limit of this series using the ratio test.
$a_n=\frac{c^n(n^2+3n+5)}{5^n(4n^n+3n+5)}$ where $c$ is a real constant
Find the limit as $n \to \infty$ of this series.
I’ve used the ratio test and written it in the form $\frac {a_{n+1}}{a_n} $
...
- 11
4
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0
answers
384
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proof-read for : equivalence Every Cauchy sequence in R converges \iff Every absolutely convergent series in \R is convergent.
I just wrote a proof to show the following statement:
Every Cauchy sequence in R converges iff Every absolutely convergent series in R is convergent.
Please let me know what you think about the proof ...
- 51
6
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0
answers
357
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
- 689
0
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2
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685
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Check the convergence of the series $\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$ and $\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$?
Check the convergence of the following series
$$\sum_{n=1}^\infty \frac{(3n-2)!!!}{3^n n!}$$ and $$\sum_{n=1}^\infty (-1)^n\frac{(3n-2)!!!}{3^n n!}$$
My attempt:
I tried Ratio test. I got
\begin{align}...
- 927
3
votes
3
answers
269
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Can we formally multiply out infinite products?
I came across Euler's proof that $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. One of the ingredients of the proof uses
$$\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)=1-\...
- 689
1
vote
1
answer
113
views
If the absolute value of an infinite series is bounded above, then the infinite series absolutely converges?
I am attempting to follow this simple example found in Further Linear Algebra by Blyth & Robertson page 14. It deduces that a sequence of partial sums must be absolutely convergent when bounded ...
- 13
2
votes
0
answers
102
views
Improper integral converges or not [duplicate]
Find all the values $\alpha \in(0,\infty)$ such that the improper integral $$\int\limits_0^\infty \frac{\Bbb dx}{1+x^{\alpha}\sin^2x}$$ is convergent.
My attempt is to analyze the cases (i) $\alpha =1$...
user1242284
1
vote
1
answer
80
views
Convergent integral
Find all the value of $\alpha >0$ such that $\int\limits_0^\infty \dfrac{\sin x}{x^\alpha +\sin x}dx$ converges.
My attempt is to check the convergence of $I_1= \int\limits_0^1 \dfrac{\sin x}{x^\...
user1240221
2
votes
0
answers
95
views
Integrability of the Jacobi Theta Function
Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that
$$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem ...
- 113
-1
votes
1
answer
78
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Absolute Convergence of a Series (maybe use ratio test) [closed]
Let
$p:=\lim_{{k \to \infty}} k(1-|\frac{a_{k+1}}{a_k}|)$
Prove
$p>1$ or $p=\infty \implies \sum_{n=1}^\infty a_n $ converge absolutely
- 962
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votes
1
answer
348
views
What happens to EX if E|X| is infinity?
---------original question----------------
According to my professor, we can divide $X$ into $X_+=\max(X,0)$ and $X_-=-\min(X,0)$, both nonnegative.
And we have $EX=EX_+-EX_-$ and $E|X|=EX_++EX_-$
For ...
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1
answer
100
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does the series $\sum | \frac{(-1)^n}{\sqrt{n}} (1+ \frac{(-1)^n}{\sqrt{n}}) |$ converge?
Does the series $$S_n = \sum \bigg{|} \frac{(-1)^n}{\sqrt{n}} \left(1+ \frac{(-1)^n}{\sqrt{n}}\right) \bigg{|}$$ converge?
I could arrive at $S_n = \sum \bigg{|} \left(\frac{1}{\sqrt{n}} + \frac{(-1)^...
- 441
0
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1
answer
168
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The Expected Value of $g(X)$
I am studying "A First Course in Probability" by Sheldon Ross, and I have come across a problem with the following proof:
Proposition 4.1
If $X$ is a discrete random variable that takes on ...
- 1,478
1
vote
1
answer
83
views
Is $A(\Bbb T) \subset C(\Bbb T)$?
My instructor defined the space $A(\Bbb T)$
as
$$A(\Bbb T) := \left\{f\in L^1(\Bbb T): \sum_{n=-\infty}^\infty |\hat f(n)| < \infty\right\}$$
and wrote $A(\Bbb T) \subset C(\Bbb T)$, where $C(\Bbb ...
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