Convergence of sequences and different modes of convergence.

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34 views

Series converges point-wise [on hold]

$$f_{n}=\sum_{n=1}^{\infty }\frac{x^{4}}{(1+x^{4})^{n}}$$ Show that it converges point-wise on $\mathbb{R}$, but not uniformly on $\mathbb{R}$. My attempt: I think, we should use Weierstrass's M ...
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1answer
25 views

Domain of $\lim_{n\to\infty}|(n+1)x| < 1$

In the process of doing the ratio test (for testing convergence of a function), I have the following issue. I am trying to find the convergence domain of $x$ in the following function: $$ ...
0
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1answer
23 views

Show the convergence of a sequence using a convergent sequence in a metric space

Let $X$ a metric space with metric $d$ and let $P \subset X$ not empty. Let $f :X \to \mathbb{R}$ a function and define $f(x) = \inf\{d(x,y): y \in P\}$. Let $(x_n)_{n \in \mathbb{N}}$ a convergent ...
-2
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0answers
19 views

Bounded, monotone, convergent [on hold]

Study if the sequence $X_n$ is bounded, monotone and convergent. If the sequence is convergent, find also its limit. $X_1 \in (0, 1), X_{n+1}=2X_n +13,n \in N$
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1answer
36 views

convergence of $f(x+y)$ to $f(x)$ for $y >$ to $0$ in $L^1$.

I have to show the following: for $f \in L^1(S^1)$ Show that the map $f(x) \to f_y = f(x+y)$ is continuous in the distance of $L^1(S^1)$. i.e. $\lim_{y \to 0} ||f_y-f||_1 = 0$ I am supposed to use ...
1
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2answers
44 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
0
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1answer
58 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...
0
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1answer
27 views

use L1-convergence to show integral convergence

Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 ...
0
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0answers
18 views

Non-linear simultaneous recurrence system

Given a non-linear, non homogeneous, discrete time recurrence system: $a_i^t = f_i(a_1^{(t-1)},a_2^{(t-1)},\ldots,a_k^{(t-1)},C_1)$, for all $i\in [k]$ where $C_1,\ldots,C_k$ are constants and each ...
0
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1answer
27 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
0
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0answers
20 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
2
votes
1answer
58 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
0
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0answers
18 views

Norm convergence of a net of operators

Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let ...
1
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0answers
48 views

Region of Convergence

I know to find the Region of Convergence you find the poles of the denominator, but I'm unsure what to do for the case where there is no denominator (for a and d) and what to do if the poles are ...
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0answers
8 views

Is $X_{k+1}=\frac{1}{N}\sum_{i=1}^N \Pi_{X_{k}^{1/2}v_i}$ globally convergent?

Let $X_0=X_0^\top\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix, let $v_i\in\mathbb{R}^n$, $i=1,\dots, n$, be a set of $n$-dimensional real vectors and pick an integer $N>0$. I ...
2
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1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
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1answer
31 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
2
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1answer
25 views

Does $\mathbb{P}$-a.s. convergence preserve independence?

Let $\mathcal F$ be a $\sigma$-algebra and $X_n$ RV s.t. $X_n$ is independent of $\mathcal F$ for all $n$. Also let $X_n \to X$ $\mathbb{P}-$a.s.. Is $X$ independent of $\mathcal F$ now too?
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26 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
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5answers
40 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...
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1answer
36 views

Determine for what values $z \in \mathbb{C}$, $\sum_{n = 1}^{\infty} \frac{z^n}{n^2}$ is convergent.

I am not sure where to start on this one. I know that $z^n$ can be written as $\sum_{n=0}^{\infty} \frac{1}{1-z}$. But I do not know how to proceed.
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2answers
29 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
6
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0answers
71 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
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2answers
25 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
6
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0answers
63 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges: $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
0
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2answers
37 views

Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
0
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1answer
39 views

Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n $ converges to $0$ for a stabilised $k\in\mathbb{N}$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for ...
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0answers
27 views

Uniform continuity of this function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. We assume that $f$ satisfies the following property: For every sequence of real numbers $(x_n)_n$, there exist a subsequence $(x_{\phi ...
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26 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
0
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3answers
38 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
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3answers
27 views

Convergent sequences out of bounded sequences

Let us consider a bounded sequence $\{a_n\}$ . Now as it is a bounded sequence it must contain a convergent sub-sequence, $\{b_n\}$. Now let us filter out $\{b_n\}$ out of $\{a_n\}$. As such we are ...
3
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2answers
33 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
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0answers
28 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
0
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2answers
58 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
0
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1answer
44 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
3
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1answer
57 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
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3answers
21 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
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0answers
48 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
5
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3answers
139 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $ if \ 0 < |x-a| < \delta$ Why can't we weaken the assumption ...
0
votes
1answer
29 views

Convergence of the sequence $z_1=\frac{3}{2}$ with $z_{n+1}=\sqrt{3z_n-2}$

Prove the convergence of the sequence $(z_n)$ such that : $$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$. Calculate also the limit. I have applied induction: $$\frac{3}{2} ...
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1answer
74 views

$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=$?

$$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=?$$ I am allowed to used all the classical techniques of calculus, and this was a question from measure theory when we were ...
5
votes
7answers
151 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
0
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1answer
33 views

Find the limit of $P_{\theta_n}\Big(\sqrt{n}(T_n-\mu(\theta_n))<z_\alpha \sigma(0)-\sqrt{n}(\mu(\theta_n)-\mu(0))\Big)$

Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow ...
0
votes
1answer
23 views

How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
2
votes
3answers
47 views

Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, $a_n = m$.

Suppose that $a_n$ is a sequence of integers. Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, $a_n = m$. I'm ...
0
votes
1answer
26 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
0
votes
1answer
17 views

Convergence of two sequences conjecture

If $(a_n) \rightarrow 0 $ and $(b_n - a_n) \rightarrow 0$, then $(b_n) \rightarrow a$. I can't think of any counterexamples to this conjecture but I'm not sure how to prove it. Any help would be ...
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vote
0answers
9 views

Monotonically decreasing sequence/series proof [duplicate]

I have a proof that I'm working on and it goes like this: Assume $a_k > 0$ and $a_k$ is monotonically decreasing. Show that: $$\sum_{k=1}^\infty a_k < \infty \iff \sum_{k=0}^\infty b_k < ...
0
votes
2answers
30 views

Finding the value for $x$ for which the series converges

I am unable to follow one if the steps described in this question. My question is, can someone describe the part I do not understand? The full question and solution is below : Find the value for $x$ ...
0
votes
0answers
27 views

Convergence of improper integral with parameter

In my assignment I have to study the convergence of this integral: $$\int_{0}^{1} \frac{ln(1 + \sqrt{x})}{x (x^{\alpha}-1)} dx$$ with the parameter $\alpha >0$. In a neighbourhood of $x=0$ I ...