Convergence of sequences and different modes of convergence.

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2
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1answer
33 views

About convergence of $(T_nR_n)$ when $(T_n),(R_n) \subset B(X)$

Let $X$ be a Banach space and $(T_n),(R_n) \subset B(X)$. (a) Prove that if $(T_n)$ converges strongly and $(R_n)$ converges strongly or uniformly, then $(T_nR_n)$ converges strongly (b) Prove that ...
3
votes
1answer
346 views

Maclaurin series for $e^x +2e^{-x}$

I'm currently stuck on the question regarding the Maclaurin series for $e^x +2e^{-x}$ I've found that the power series representation for it is $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
1
vote
1answer
76 views

Fixed point iteration for $\sqrt[3]{a}$

So I'm given the scheme for computing $\sqrt[3]{a}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this scheme is as fast as possible. Any hints ...
2
votes
1answer
49 views

Limit of double integral from kernel converges to zero

Let $g_t(x)=\dfrac{1}{2\sqrt{\pi t}}e^{-\frac{(at+x)^2}{4t}} $ and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow ...
-2
votes
1answer
99 views

Convergence on sum of cos [closed]

How to find the range of x on this sum to converge? $$\sum_{n=1}^∞{{\cos nx}\over{n}}.$$
1
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1answer
98 views

Prove that $ (\sqrt{a_{n}})_{n=1}^{\infty}\rightarrow \sqrt{a} $ when $ (a_{n})_{n=1}^{\infty}\rightarrow a $

I'm stuck on a homework question, and would be very thankful to anyone who helps. The question: Prove or disprove the following statement: If: $ (a_{n})_{n=1}^{\infty} $ is a series of positive ...
4
votes
1answer
42 views

Limit outside compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $1\leq p<\infty$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)\textrm dx=1$. Define ...
1
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1answer
52 views

Convergence of a sequence in a Hilbert space

My problem is the following : Let $\mathcal{V}$ be a Hilbert space and ( . ) the corresponding scalar product. Let $a_1,\cdots,a_M$ be $M$ linearly independent elements of $\mathcal{V}$. Let ...
0
votes
1answer
89 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
4
votes
1answer
33 views

Supremum over compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ It follows ...
1
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2answers
34 views

Need a hint for sequence convergence homework

Hello and thank you for spending time to help me in advance! The following exercise/homework problem has been giving me a difficult time and I am wondering if there is something simple that I am just ...
3
votes
1answer
105 views

$X_{n}$ converges to $X$ in distribution iff $E\{f(X_{n})\} \to E\{f(X)\}$ for all bounded $ f \in C^{\infty}$.

Let $(X_{n})_{n\geq 1}$, $X$ be $\mathbb{R}$-valued random variables. Show that $X_{n}$ converges to $X$ in distribution iff $E\{f(X_{n})\}$ converges to $ E\{f(X)\}$ for all bounded $C^{\infty}$ ...
2
votes
1answer
36 views

Does $T_1$ imply Fréchet–Urysohn (every limit point is a limit of some sequence)?

It's not a problem from a book so I’m not even sure the statement is true. Nevertheless here's an alleged proof: ADDED LATER: Although the result is wrong I can't find a problem with the proof. I ...
0
votes
2answers
103 views

Show that $\sum\limits^\infty_{k=1}k^{-s}$ converges if and only if $s>1$ for positive $s$.

Show that $\sum\limits^\infty_{k=1}k^{-s}$ converges if and only if $s>1$ for positive $s$. Sorry, no thinking process so far. I have a blockage right now. Any hint is much appreciated.
1
vote
1answer
58 views

Rate of convergence in an infinite geometric series of matrices

I have the following system $Z=[A^t + A^{t-1}+A^{t-2}+....+I]*E$, in which $A$ is a $n\times n$ matrix and $Z$ and $E$ are $n\times1$ vectors. The eigenvalues of $A$ are all smaller than one and the ...
7
votes
3answers
208 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) ...
5
votes
3answers
324 views

Evaluate $ \sum\limits_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}$

The question was: Evaluate, ${\textstyle {\displaystyle \sum_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}}}.$ And I go, since $\frac{n}{n^{4}+n^{2}+1}\sim\frac{1}{n^{3}}$ and we know that ${\displaystyle ...
0
votes
1answer
48 views

Proving Series Convergence Using Taylor Polinomials

I ask for some help with this question: Suppose f(x) is continuous function on [0,1] and twice differentiated at x=0, and $u_n=(-1)^n f\left(\frac 1n\right)$. Prove : if $f(0)=0$ then ...
3
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1answer
154 views

Show that $(X_{n},Y) \to^{\mathcal{D}} (X,Y)$ AND if $X=h(Y)$ where $h$ is a Borel function that $X_{n}\to^{P} X$

Let $X_{n}$, $X$, and $Y$ be real-valued r.v.'s all defined on the same space $(\Omega, \mathcal{A},\mathbb P)$. Assume that $\lim_{n \to \infty}\mathbb E\{f(X_{n})g(Y)\}=\mathbb E\{f(X)g(Y)\}$ ...
1
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2answers
25 views

Series, conditional or absolute? +more

This is for a presentation, so I not just want to solve it, but also be able to talk a bit about it $$\sum \limits_{n=2}^\infty (-1)^{n+1}\frac{n-1}{n^2}$$ 1)Show that the series converges, is it ...
1
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1answer
69 views

verify, that this sequence is cauchy-sequence

Be $ 0 < q < 1, (a_n)_{n \in \mathbb{N}} $ a sequence in $\mathbb{R}$ and $ n_0 \in \mathbb{N} $ One has: $ |a_{n+1} -a_n| \leq q |a_n-a_{n-1}|$ for all $n \geq n_o$. Show, that the ...
2
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2answers
137 views

Showing that a sequence of random variables with $\mathcal{L}(X_{n})$ (I.e., law) uniform on $[-n,n]$ does not converge at all

Let $(X_{n})_{n\geq 1}$ be a sequence of real valued random variables with $\mathcal{L}(X_{n})$ (that is, law or distribution) uniform on $[-n,n]$. In what sense(s) do $X_{n}$ converge to a random ...
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0answers
351 views

Clarification of “Convergence almost everywhere implies convergence in measure”

I have looked over the proofs that show convergence a.e. imply convergence in measure. I understand the proofs, but I do not understand why one must go into such detail. It seems as though one ...
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3answers
253 views

Find the radius of convergence and the interval of convergence of the series $\sum_{n=1}^\infty \frac{n(x-4)^n}{n^3+1}$

Series is: $$\sum_{n=1}^\infty \frac{n(x-4)^n}{n^3+1}$$ So, I understand that I use the ratio test to find r, but I can't simplify the equation to the point where I can do this. Here's where I am so ...
6
votes
1answer
129 views

What number does $\sum_{k=1}^{\infty}\ln^k(1+\frac{1}{k})$ converge to?

What number does$$\sum_{k=1}^{\infty}\ln^k(1+\frac{1}{k})$$converge to? I think it converges by root ...
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votes
2answers
60 views

Convergence $ a_1 = 1, \quad a_{n + 1 } = \sqrt{2a_n} \quad n\in N $ [closed]

I am stucked at this exercise.. Is this sequence convergent? If yes, what the is the limit? $ a_1 = 1$ and $ a_{n + 1 } = \sqrt{2a_n} \quad n\in N $
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1answer
278 views

Connection between almost sure convergence and lim sup and lim inf

I noticed other similar questions about this, but I'm going in circles. Am I correct that the following are equivalent ways of showing $X_n$ a sequence of random variables converges almost surely to ...
3
votes
1answer
98 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
1
vote
1answer
41 views

Convolution convergent in $L^p$

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
2
votes
2answers
69 views

If $|X_{n}| \leq Y$ almost surely, show that $\sup_{n}|X_{n}|\leq Y$ almost surely as well.

Suppose $|X_{n}|\leq Y$ a.s., each $n$, $n=1,2,3,\cdots$. Show that $\sup_{n}|X_{n}|\leq Y$ a.s. also. This seems pretty intuitive to me, since if $|X_{n}|\leq Y$ a.s., it is bounded above by $Y$, ...
1
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1answer
87 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
2
votes
3answers
100 views

Is series convergent/divergent

I need to find out is series convergent or not $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ How can I do that? Can you show step-by step solution?
2
votes
1answer
71 views

Convolution converging uniformly on real line

I'm working on this question and stuck with the following part: Suppose $f\in L^\infty(\mathbb{R})$ and $K,K_1,K_2,\ldots\in L^1(\mathbb{R})$ with $K_n\rightarrow K$ in $L^1$. Why is it true that ...
0
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3answers
43 views

Convergent series: am I doing it right?

I have $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ I get (i did not write all solution as it is quite hard to me to put this in LaTeX by myself) $$ \lim_{k\rightarrow \infty} ...
2
votes
0answers
60 views

Convergence a.s. implies convergence in $L^{1}$

Suppose $\lim_{n \to \infty}X_{n} = X$ almost surely. Let $Y=\sup _{n}|X_{n}-X|$. Show that $Y < \infty$ almost surely, and define a new probability measure $Q$ by ...
2
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0answers
34 views

Radius of Convergence and Interval of Convergence: Am I doing it right?

I must find ROC and IOC in $$ \sum_{n=1}^\infty \frac{{(-1)}^n(x^{2n})}{\sqrt[3]{n^2+4n}} $$ I get $$ R= \lim_{n\rightarrow \infty}\left| \frac{a_n}{a_{n+1}} \right| = \cdots = \lim_{n\rightarrow ...
1
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1answer
63 views

Using a comparison test to see if a series converges or diverges

The series is a sum of $(n^2+1)/(n^3+2)$ to infinity from $n=1$. From previous questions I have seen that you should try to alter the sum to one that is commonly known so a comparison can be made. I ...
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1answer
128 views

A strange (seemingly pointless) exercise on convergence of series

I have come across an exercise which asks to prove that the series of functions$$\sum\frac{x^n}{1+x^n}$$ is convergent for $x\in [0,1)$. It also asks us to prove that the series converges uniformly ...
0
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1answer
45 views

What would be the simplified form of this expression?

I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows: $$ M_{X_n}(ju)= ...
1
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1answer
120 views

Show $\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ converges uniformly on $\mathbb{R}$

$\sum^\infty_{n=1}\frac{x}{n^{0.6}(1+nx^2)}$ converges uniformly on $\mathbb{R}$ Is $x\rightarrow\sum^\infty_{n=1}(\frac{x}{n^{0.6}(1+nx^2)})$ continuous at all points of $\mathbb{R}$? I'm stuck on ...
2
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3answers
126 views

How can I determine convergence for this equation: $\sum_{n=1}^\infty (-1)^{n+1}\frac{n}{n^2+4}$

Again, my equation is: $$ \sum_{n=1}^\infty (-1)^{n+1}\frac{n}{n^2+4} $$ I believe I'm supposed to use a basic comparison check, so am I able to ignore the -1 portion at the start of the equation? ...
0
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3answers
83 views

How to show that $\sum {2^j + j \over 3^j - j}$ converges

I'm not quite sure how to go about growing the numerator or shrinking the denominator to perform a tricky comparison test, or hashing out the ratio test, so any help would be much appreciated!
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0answers
46 views

Finding rate of convergence

I'm trying to determine a rate of convergence for a non linear function $f(x)=x^5 + 12x^3 -130$ to find its root. Using the fixed point iteration, I am using the second form function $g(x)=(4x^5 + ...
0
votes
1answer
36 views

The weak law of great numbers from the central limit theorem.

I couldn't derive the weak law of large numbers from the central limit theorem for iid random variables with $0 < \operatorname{Var}(X) < \infty$. The central limit theorem gives $$\frac{\sum ...
1
vote
1answer
31 views

Using the comparison Test to see if a series converges or diverges

∑1/(1+(n^3)): Im trying to use the limit comparison test, but I'm struggling to find the comparing equation. I would appreciate if someone could either give me advice to finding the comparing ...
0
votes
5answers
101 views

Is series convergent?

Does this series converge? $$\sum_{n=1}^{\infty }\frac{{(-1)}^{n}(n+2)}{{n}^{2}+4}$$ How can i step-by-step calculate it?
0
votes
1answer
43 views

checkig for convergece (TIFR GS $2010$)

Question is to check for convergence of sequence $(x_n)$ defined as : $x_1=0.1,x_2=0.101,x_3=0.101001$,....... What i have done so far is : I could not guess what would be $x_4$ but tried to write ...
2
votes
1answer
70 views

Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots+ b_n)$ convergent?

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = 1/(na_n^2)$ for $n\ge1$. Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots + b_n)$ ...
1
vote
1answer
885 views

Convergence in $L^p$ norm implies pointwise convergence almost everywhere? [duplicate]

Fix a real number $1\leq p<\infty$. Is it true that if functions $f\in L^p(\mathbb{R})$ and $f_1,f_2,\ldots\in L^p(\mathbb{R})$ are such that $\|f_n-f\|_p\rightarrow 0$ as $n\rightarrow \infty$, ...
5
votes
2answers
272 views

Does the series $\sum_{n=2}^\infty \frac {\sin(n+\frac 1n)}{(\ln n)^2}$ converge?

Could you give me some hint how to deal with this series ? I could not conclude about absolute convergence because $\frac {\left|\sin\left( n+\frac 1n \right)\right|}{(\ln n)^2}\le \frac 1{(\ln ...