Tagged Questions

Convergence of sequences and different modes of convergence.

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1
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1answer
33 views

How can I prove that Xn converges to 0 in probability?

Let $X_n\sim U[-1/n,1/n]$. Since for convergence in probability for every $\epsilon>0$, $$ \lim_{n\to\infty} P(|X_n - X|\ge \epsilon) = 0 $$ Hence, $P(|X_n-0|\ge ...
0
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2answers
42 views

show that sequence $(a_n)$ is convergent [closed]

Let $(a_n)$ be a sequence of real numbers. If $$|a_{n+1} − a_n | ≤ \frac{1}{2}\ |a_n − a_{n−1} |, \qquad \forall n \in \{2, 3, \dots\}$$ then show that $(a_n)$ is convergent.
0
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1answer
35 views

Convergence of Real Series

Consider the following real series: $$g_k(x)= \left( \frac{(-1)^k}{\sqrt{k}} \right ) \cos kx$$ on $\mathbb{R}$. Does the series converge pointwise or uniformly? Check the continuity of the limit ...
4
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0answers
64 views

Methods of constructing rapidly convergent series

It's fairly easy to see that the series $$1-\tfrac{1}{3}+\tfrac{1}{5}-\cdots=\tfrac{1}{4}\pi$$ is : 1. Convergent to the value given, and - 2. Very slowly converging, which can be seen just by ...
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2answers
66 views

Determining whether $ \displaystyle \sum_{n = 0}^{\infty} 4\cos(2\pi n)e^{-3n} $ diverges

Consider the following infinite series: $$ \displaystyle \sum_{n = 0}^{\infty} 4\cos(2\pi n)e^{-3n} $$ Determine whether the infinite series diverges or converges. I tried to use: The Integral ...
0
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1answer
25 views

A conjecture concerning convergence of a kind of recursive sequence

Along the thread given here, Find the limit of a recursive sequence, I am very curious about the limit of the recursive sequence defined by \begin{gather*} u_0>0, \quad u_1>0,\quad ...
1
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2answers
78 views

Show $\{x_n\}$ approaches zero “faster” than $1/n$

Prove that if $\{x_n\}$ is a monotone decreasing sequence of positive numbers and $\sum_{n=1}^\infty x_n$ converges, then $\lim_{n\to\infty}nx_n=0$, i.e., $x_n$ approaches zero "faster" than $1/n$.
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5answers
87 views

Prove the following sequence is a Cauchy Sequence

Let $\{X_n\}$ be the sequence defined recursively by $x_1=2$ and $x_{(n+1)}=(x_n/2)+(5/x_n)$. Prove that $\{x_n\}$ converges and find the limit of the sequence. I understand the definition of a ...
0
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1answer
46 views

Convergence of $\int_0^\infty\frac{\sqrt{x+\sqrt{x^{-1}+x}}}{\sqrt{x\phantom{|}}\sqrt{x^{-1}+x}}dx$

I've been trying to evauluate this integral $$\int_0^\infty\frac{\sqrt{x+\sqrt{x^{-1}+x}}}{\sqrt{x\phantom{|}}\sqrt{x^{-1}+x}}dx$$ So far I have tried the substitution $$t =1/x +x$$ But this has made ...
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2answers
23 views

Estimate for Dominated Convergence

I'm trying to check that there is an estimate: $$n|e^{ia/n}-1|\leq C\left(|ia|+1\right)\quad(n\in\mathbb{N})$$ for some constant $C\geq0$ and all reals $a\in\mathbb{R}$. I had some tries but just ...
0
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1answer
9 views

ratio through convergence topologies

If $X$ is sapce topological and two topologies $\tau_1$, $\tau_2$ both metrizable D1, D2 (metric). If convergence with $D_1$ implies convergence with $D_2$, then It might from that condition, it ...
0
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1answer
32 views

How can I prove that Xn converges to 0 in distribution?

Xn~U[-1/n,1/n]. Since for convergence in distribution Xn-->X iff Fn(x)-->F(x). First of all, I am trying to get the cumulative and it is Fn(x)=(1+xn)/2. Hence, am I doing something wrong?
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1answer
25 views

Notion of convergence of equivalent norms is the same

I would like to make clear the proof for the following theorem which states that two norms over a vector space are equivalent iff their notion of convergence is the same. I have an hint for the proof ...
6
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1answer
254 views

Calculate the infinite sum $\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)…(k+p)} $

I have to prove that $$\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)....(k+p)} $$ is equal to $\, \dfrac{1}{(\,p-1)!p}.$ How can I do that?
2
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1answer
90 views

How does convergence in total variation of random variables compare to other modes of convergence?

Suppose $X,Y$ are random variables. We define the total variation distance of random variables to be $$d_{TV}(X,Y)=\inf \lbrace\mathbb{P}(|X'-Y'|>0): \text{ $X', Y'$ are couplings of $X,Y$ ...
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1answer
37 views

Convergence of continuously differentiable functions on a compact interval.

I'm trying to figure out this excercise about convergence and could really use some help. I have partial answers that I'd like for you to check and would really appreciate hints for the others. Let ...
0
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1answer
14 views

To show that Subsequence converges uniformly

Let {$f_n$} be a uniformly bounded sequence of functions which are Riemann integrable on $[0,1]$ and $$F_n (x) = \int_{0}^{x} f_n(t)dt \hspace{0.2in} ( 0 \leq x \leq 1 ) $$ I need to prove that there ...
0
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0answers
16 views

convergence series functional and numerical series

I have this question: Can a covergent numerical series be a convergent functional series?I know the other way that a functional series can be a numerical series because i take functions as a cnstant ...
2
votes
1answer
57 views

Find the limit of a recursive sequence

Let $(u_n)_n$ be a real sequence such that $$ u_{n+2}=\sqrt{u_{n+1}}+\sqrt{u_{n}},\,u_0>0,\,u_1>0. $$ Fisrt, it is easy to check that $(u_n)_n$ is well defined and $u_n>0$ for all ...
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5answers
48 views

Convergence of series with ^(n+1)

I want to check, if this series is convergent or not: $$\sum\limits_{n=1}^\infty \frac{n^n(n!)}{(2n)!}$$ I tried with the ratio test: ...
1
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2answers
28 views

Sum convergence

I want to check this sum: $$\sum\limits_{n=1}^\infty (\frac{(3n+1)!}{n!(2n+1)!}*7^{-n})$$ I think, the easiest way is to use the ratio test: ...
1
vote
1answer
41 views

Pointwise limit of a sequence of continuous functions is discontinuous at most finitely/countably many points.

Let $\{f_{n}\}$ be a sequence of functions in $C[0,1]$ such that $f_{n}\to f$ pointwise. Then $f$ has at most finitely (or probably countably) many discontinuities. Is this statement TRUE or FALSE? ...
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1answer
261 views

Sums in $\mathbb N^3$

Assume that $a_{n,m,k}\geq 0$ is a sequence, $n,m,k\in \mathbb N$, such that $$\sum_{n,m,k\in \mathbb N} a_{n,m,k}^2 <\infty$$ i.e. it is in $\ell^2 (\mathbb N^3)$. I want to prove the following: ...
3
votes
1answer
52 views

Does this simple sum converge

I'm trying to determine whether the sum $$S=\frac{2}{1}+\frac{2\cdot 5}{1\cdot 5}+\frac{2\cdot 5\cdot 8}{1\cdot 5\cdot 9}+...+\frac{2\cdot 5\cdot 8...(3n-1)}{1\cdot 5\cdot 9...(4n-3)}+...$$ converges ...
0
votes
0answers
18 views

convergence for dependent random variables

Let $X_1,X_2,\ldots,X_n$ be a sample of iid rvs. Suppose their distribution has a parameter $0<\theta<\infty$ and support $(0,\infty)$. Let $\theta_n=\theta_n(X_1,\ldots,X_n)$ be an estimator of ...
0
votes
1answer
76 views

Show that the hyperplane $H=\{x \in\mathbf{R} \;|\; : {A}^{T} x =b\}$ is closed in $\mathbf{R}^d$.

Let $A \in \mathbf{R}^d \backslash \{0\}$ and $b\in\mathbf{R}$. Show that the hyperplane $H=\{x \in\mathbf{R} \;|\; : {A}^{T} x =b\}$ is closed in $\mathbf{R}^d$.
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1answer
39 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 ...
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2answers
105 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
18 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
27 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
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1answer
26 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
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1answer
27 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
19 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
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2answers
23 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 ...
9
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1answer
451 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 ...
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4answers
68 views

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

How to show that, for any fixed constant $a\in(0,1)$, the series $$ \sum_{n=1}^{\infty}(-1)^n \sin\left(\frac{a}{n}\right). $$ is convergent yet not absolutely convergent. My idea is to express ...
0
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1answer
47 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
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2answers
32 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 ...
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2answers
78 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
votes
0answers
21 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
69 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: ...
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2answers
25 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
vote
3answers
54 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... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
1
vote
1answer
16 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
vote
4answers
43 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
21 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
64 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
32 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
19 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 ...