Tagged Questions

Convergence of sequences and different modes of convergence.

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0
votes
1answer
24 views

What about the convergence of the geometric mean sequence of the terms of a given convergent sequence?

Let $(a_n)$ be a sequence of positive real numbers such that $$ \lim_{n\to} a_n = a.$$ Let $b_n \colon= \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdots a_n}$ and $c_n \colon= a_n^{a_n}$. Then what can we say ...
1
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2answers
36 views

How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$?

Let $\mathbb{R}$ be the set of all real numbers under the usual metric $d$ defined as follows: $$d(x,y) \colon= |x-y|$$ for all $x$, $y$ in $\mathbb{R}$, and let $\mathbb{Q}$ be the set of all ...
0
votes
1answer
10 views

Is the monotone convergence theorem bidirectional?

Say I have $(f_n)$ with $f_1 \le f_2 \le ...$ and I know that $\lim_n\int f_n<\infty$ exists, does that imply $f_n$ converges a.e.? Most formulations I have seen of the monotone convergence ...
2
votes
1answer
14 views

Supremum of the function of a sum for Weierstrass M-test

I have to prove the uniform convergence of $\sum_{k=1}^\infty \frac{k+z}{k^3 + 1}$ on the closed disc $D_1(0)$. Using the M-test, $|\frac{k+z}{k^3 +1}| \leq |\frac{k+1}{k^3 +1}| = \frac{1}{k^2 - k + ...
1
vote
1answer
20 views

convergence of a series..

This might be ridiculously easy but I just forgot about series. Consider the series $\sum_{k=1}^\infty \frac{1}{k^2-2}$. Does it converge? What about $\sum_{k=1}^\infty \frac{1}{k^2-r}$ for any ...
1
vote
1answer
20 views

Speed of convergence in probability

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $E(X_i)=\mu$. Let $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^{n}X_i$. Let $\{A_n\}_{n \in ...
0
votes
1answer
10 views

How can I complete my proof: Sobolev space W^(1,p) is complete? Using Convergence theorem

I'm trying to prove that W^(1,k) (R) is complete. The steps i Had so far: let {fn} be a cauchy sequence in W^(1,k). therefore {fn} and {dfn} are cauchy sequences in L^p(R), and therefore converge ...
0
votes
2answers
21 views

Relations among notions of convergence

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Does this imply that $plim_{n\rightarrow \infty}A_n=0$, where $plim$ is the probability ...
3
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0answers
279 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function (not necessarily continuous). Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. 1. Let ...
0
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4answers
56 views

How to show that $\sum_{n=1}^{\infty}(-1)^n \sin(a/n)$ converges but not absolutely.

How to show that, for any fixed constant $a\in(0,1)$, the series $$\sum_{n=1}^{\infty}(-1)^n \sin\frac{a}{n}$$ is convergent yet not absolutely convergent. My idea is to express sin(x) as series but ...
0
votes
1answer
32 views

Application of Slutsky's Theorem

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$. Let ...
0
votes
2answers
28 views

$\sum{\frac{1}{\sqrt{4n+1}}+\frac{1}{\sqrt{4n+3}}-\frac{a}{\sqrt{2n}}}$ converges?

Can we find a constant $a$ such that $\sum{\frac{1}{\sqrt{4n+1}}+\frac{1}{\sqrt{4n+3}}-\frac{a}{\sqrt{2n}}}$ converges? Try: I am trying to compare the n th term with $\frac{c}{\sqrt{n}}$ where c is ...
1
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2answers
63 views

Question regarding convergent series

If the series with general term $a_n^2$ converges, why does the series with general term $a_n/n$ converge as well??? A peer of mine showed me this, but I really don't find it obvious and I really ...
0
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0answers
16 views

convergence of 2 series in the critical strip

let us define 2 series: $$A=\sum_{k=1}^{+\infty}(-1)^{(2k+1)}\frac{\ln(2k+1)}{(2k+1)^s}$$ $$B=\sum_{k=1}^{+\infty}\frac{\ln(2k)}{(2k)^s}$$ Define $$ s=\alpha + \beta i$$ Does $\frac{A}{B}$ go to ...
6
votes
2answers
64 views

Find $S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+…+\frac{2n-1}{2^n}+…$

I'm trying to calculate $S$ where $$S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+...+\frac{2n-1}{2^n}+...$$ I know that the answer is $3$, and I also know "the idea" of how to get to the ...
2
votes
2answers
33 views

Limit Computation, Sandwich.

I have the following question. I was asked to compute the following limit: Let $A_1 ... A_k$ be positive numbers, does exist: $$ \lim_{n \rightarrow \infty} (A_1^n + ... A_k^n)^{1/n} $$ My work: ...
1
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2answers
22 views

How to determine if this converges?

I find this one to be hard, I tried expanding $\cos$ into a Taylor series, but that didn't work out well because I couldn't apply $p$-series... $$ \sum^{\infty}_{n=1} n (1-\cos(\pi/n)) $$
1
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3answers
42 views

How to determine if this series converges?

Does this series converge? I tried using limit comparison, and I don't know what to try next... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
0
votes
1answer
13 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
1
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4answers
37 views

Convergent complex series

Is $$\sum\limits_{n=1}^\infty \frac{i^n}{n} $$ convergent? Im confused as to how to solve this question, I've been trying to use ratio test but that doesn't seem to be helping.
1
vote
1answer
20 views

convergence and nested logs

The problem is to test convergence for the series: $\sum^\infty_{n=3}1/(\ln n)^{\ln(\ln(n))}$ I tried manipulating the log term (by means of ...
5
votes
3answers
57 views

Divergence of $a_{n+1}=\sqrt{2a_n+3}$?

I am wondering what I am missing from my proof. I would like to show that the limit of the sequence $$a_{n+1}=\sqrt{2a_n+3},\,\, a_1=4,$$ goes to $\infty$, as $n \rightarrow \infty$. Is there ...
3
votes
2answers
30 views

Test for convergence for $\ln \frac{n^2}{n^2-1}$

I've tried to figure out if this converges using the comparison test, and the ratio test, but with no luck: $\sum^\infty_{n=2} \ln(n^2/(n^2-1))$. I'd appreciate any help
1
vote
1answer
15 views

Understanding a proof that bounded sequences in $\mathbb{R}^p$ has a convergent subsequence

I'm having trouble concerning the following proof that each bounded sequence in $\mathbb{R}^p$ has a convergent subsequence. We have already established that this is true in $\mathbb{R}$ and this is ...
0
votes
1answer
22 views

Applying the monotone convergence theorem

Recently learned about the monotone convergence theorem. I have the sequence: $x_n = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}$ I need help proving that it is increasing and bounded, ...
2
votes
2answers
59 views

$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n} = e^{-\frac{x^2}{6}} $

I am wondering about a limit that wolframalpha got me and that you can find here wolframalpha It says that $$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} ...
1
vote
2answers
51 views

Convergence of a series with positive terms

Let $(a_n)_n$ be a strictly positive sequence . How to prove that the series $$ \sum\limits_{n = 1}^\infty {\frac{{a_n }}{{(a_1 + \cdots + a_n )^2 }}} $$ converges ? Any ideas ?
2
votes
1answer
60 views

integral over $\sin(x)/(x^p)$ from 0 to $\infty$ [on hold]

I have a question about the convergence of $\int^\infty_0 \frac{\sin(x)}{x^p} dx$ for $p\in\mathbb{R}$ what can I say for convergence of this function ?? I know that $\int^\infty_0 ...
-2
votes
0answers
9 views

Functional row of complex variable [on hold]

I got a problem with convergence of this row (the first one)
0
votes
1answer
14 views

The converge of expectation value based on almost sure convergence

Here is the question: Let $\xi_n $ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P) $ such that $E \xi_n^2 \le c $ for some constant $c$. Assume that $\xi_n \to \xi ...
0
votes
0answers
11 views

Convergence in distribution of a serie

How could we prove that this serie converge in distribution to a centered gaussian variable ? $$ \frac{1}{\sqrt{n^3}} \sum_{i,j,k = 1}^{n} x_{i,j} x_{j,k} x_{k,i} $$ with for all $ i,j \in ...
2
votes
3answers
58 views

Is the following series converging $\sum_{n=2}\dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Is the following series converging $\sum_{n=2}\dfrac{1}{(\ln n)^{\ln n}}$ I am not able to compare this with anything, can some show the way
1
vote
2answers
53 views

About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
2
votes
1answer
32 views

Is the limit of a sequence of random variables unique?

If $Y$ and $Z$ are two distinct random variables with the same distribution (for example maybe $Y$ is constant equal to $1$ and $Z$ is equal to $1$ almost everywhere), then surely any sequence $X_n$ ...
2
votes
1answer
30 views

Convergence of $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$

Let $(x_n)$ be the sequence defined by $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$ for $n \geq 2$ and $x_1$ be any real number. Then I want to prove that $x_n$ is convergent. It is ...
1
vote
1answer
25 views

Convergent Sequences

Is the sequence $\{$cos$(\pi\sqrt{n^2+n})\}_{n=1}^\infty$ convergent? I guess it is divergent because cos function is oscillating. But not sure. I am stuck in doing justification too. Any ideas?
0
votes
1answer
22 views

Proving sequence convergence

I'm pretty confused...I understand bits and pieces but not how it all comes together...I would appreciate some help, either a written out example (you can make up one) and/or comments on how to fix my ...
0
votes
1answer
28 views

Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.

Help please! Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge. I can prove that $\sum a_n b_n$ converges but couldn't for $\sum \sqrt{a_n b_n}$. Thank you.
0
votes
3answers
18 views

Prove from definition of convergence that (-2n+5)/(3n+1) is convergent.

Prove directly from the definition of convergence that (-2n+5)/(3n+1) is convergent. So I let epsilon>0; there exists N in the natural numbers such that n greater/equal to N implies that the ...
0
votes
2answers
30 views

subsequences and convergence for real analysis

Show there is a sequence $a_n$ such that for every real number x, there is a subsequence of $a_n$ converging to x. I have a hint that is to start with a bijection a:$\Bbb N$ $\to$ $\Bbb Q$.
2
votes
1answer
28 views

Squence of order statistic converges to median?

Suppose that $(X_1,...,X_N)$ be $N$ iid samples from uniform distribution. Let $X_{(n)}$ be $n$-th order statistic. It sounds natural that $X_{N/2}$ converges to the median, $1/2$. (interpret $N/2$ as ...
1
vote
1answer
43 views

When does $\sum\limits_{n=1}^\infty \frac{p(p+1)\cdots(p+n-1)}{n!n^q}$ converge/diverge?

When does $$\sum_{n=1}^\infty \frac{p(p+1)\cdots(p+n-1)}{n!n^q}$$ converge/diverge? Please note that $p$ may be negative. Thanks!
1
vote
1answer
36 views

Every sequence with $\lim x_n=c$, show that $f$ is continuous at $c$

Let $f:S\to \mathbb{R}$ be a function and $c \in S$, such that for every sequence ${x_n} \in S$ with $\lim x_n=c$, the sequence ${f(x_n)}$ converges. Show that $f$ is continuous at $c$.
0
votes
2answers
23 views

Cubic convergence of itearative method

thank you for your time at first! It's my homework, so I don't expect answer with result, only some hint. With given iteration method $$x_{n+1} = \frac{x_n(x_n^2 + 3U)}{3x_n^2 + U} $$ show cubic ...
0
votes
0answers
11 views

Convergence in distribution to a constant

Does Convergence is distribution to a constant mean $$ \lim_{n\rightarrow \infty}F_{n} (X) = a $$ and is that equivalent to writing $P(X<x) = 0$, if $x<a$ $P(X<x) = 1$, if $x\geq a$?
0
votes
3answers
27 views

Convergence of two nested sequences

For the two following sequences I want to find their limits: (1) The sequence $2$, $2\sqrt{2}$,$2\sqrt{2\sqrt{2}}$,... (2) $a_{n+1}$ = $\sqrt{1+a_n}$, $a_1 = 1$ For both sequences I want to show ...
1
vote
3answers
23 views

Sequence which converges pointwise but not uniformly?

it might be simple but I don't find a sequence $f_n: [0,1] \rightarrow \mathbb{R}, n \in \mathbb{N}$ that converges pointwise but not uniformly. First I thought it could be $f_n(x) = \frac{x}{n}$ but ...
0
votes
3answers
72 views

Limit of a sequence $\sqrt[n]{3} $

Using a definition of limit of a sequence and Bernoulli's inequality proof that limit of $\sqrt[n]{3} $ is 1. From the definition I know that ∀ε>0 ∃N∈ℕ ∀n>N |an-g|<ε and g=1. ...
1
vote
1answer
23 views

Convergence in distribution of the Sum Y/\sqrt{\lambda}

The question follows: ${[X_n]}_{n\geq1}$ is a sequence of independent rv's such that: $P(X_n=-1)=1/2$ $P(X_n=1)=1/2$ Let $N \in Po(\lambda)$ where N is independent of ${[X_n]}_{n\geq1}$. ...
3
votes
3answers
68 views

Does $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$ converge absolutely, conditionally, or diverge?

I've been given the hint to use the binomial theorem and show that $e-\left(1+\frac{1}{n}\right)^n > \frac{1}{2n}$ for $n \geq 2$. So I've written \begin{align*} e-\left(1+\frac{1}{n}\right)^n = e ...