Convergence of sequences and different modes of convergence.

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20 views

Does $\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty$ imply $\frac{1}{n^2} \sum\limits_{k=1}^n a_k^2 \to 0$?

I'm currently working on some problem regarding Dirichlet forms and, as the title states, I'm trying to figure out if $$\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty \Rightarrow ...
0
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5answers
46 views

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge.

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge. Can some explain how to approach this problem. All Ideas are appreciated
0
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2answers
33 views

trouble solving this sequence problem

I'm having some trouble solving this problem about sequences: a(n): a(1) = 2; a(n+1) = (a(n) + 1)/2, n belongs to N(natural numbers) 1)Prove that this sequence is monotonically decreasing 2)Prove ...
4
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4answers
45 views

Limit of $(1-2^{-x})^x$

I am observing that $(1-2^{-x})^x \to 1$ as $x \to \infty$, but am having trouble proving this. Why does the $-x$ "beat" the $x$? I thought of maybe considering that $$1-(1-2^{-n})^n = ...
0
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2answers
38 views

How to properly state as to why $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges.

So I know that $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges, because the highest power in the numerator is $n^\frac{3}{2}$ and the highest power in the numerator is $n^4$, so I have ...
1
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2answers
40 views

How to determine whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges.

I am trying to find whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges. I used the limit test, and it comes out as inconclusive since ...
3
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0answers
26 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
2
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1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
0
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1answer
32 views

Calculus sequences And series

Find the values of $x$ for which the series $\sum_o^\infty \frac {(x+3)^n}{2^n}$ converges. I took it as $(\frac {x+3}2)^n$ then used the rule of summation of $r^n= \frac 1{1-r}$ then found ...
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0answers
37 views

Can this convergent series be generalised?

A friend of mine gave this question,I have no idea how to even start generalising the nth term of the series so that I can summify it to n tending to infinity. $$\frac{1}{(1!)} ...
1
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1answer
13 views

Joint convergence in distribution

I've one question concerning convergence in distribution of random variables: Let $X_n \rightarrow X$ and $Y_n \rightarrow Y$ for $n \to \infty$ where $\rightarrow$ denotes convergence in ...
1
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0answers
26 views

How i could show that this inequality true or false: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$?

Is this inequality true: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$ ? note :$s=\sigma + it$, where $\sigma, t\in \mathbb{R}$. I would be interest for any replies ...
2
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3answers
59 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{āˆž} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{āˆž} \frac{\ln n}{n^2}$ to ...
1
vote
3answers
20 views

For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?

For real valued function $f$ define $$S(f)=\{x:x>0,f(x)=x\}$$ For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? $\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan ...
1
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1answer
53 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{nā†’\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
1
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2answers
32 views

convergence of $\sum_n x_n$ versus $\sum_n f(x_n)$ for differentiable $f$

Suppose $f(0) = 0$, $f$ is differentiable at 0, $f'(0) \ne 0$, and $x_n \rightarrow 0$. What can we say about (i) the convergence of $\sum_n x_n$ versus (ii) the convergence of $\sum_n f(x_n)$? It ...
0
votes
1answer
22 views

Absolute Convergene of the Product Series

Theorem Suppose (a) $\sum_{n=0}^{\infty}a_n$ converges absolutely, (b) $\sum_{n=0}^{\infty}a_n=A$, (c)$\sum_{n=0}^{\infty}b_n=B$, (d)$c_n=\sum_{k=0}^{n}a_kb_{n-k}$ $(n=0,1,2,\dots)$. Then ...
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3answers
50 views

Question about $e^x$

Let $ p(x)=1+x+x^2/2!+x^3/3!+....+x^n/n!$ where $n$ is a large positive integer.Can it be concluded that $\lim_{x\rightarrow \infty }e^x/p(x)=1$?
2
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3answers
126 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...
0
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1answer
10 views

Convergence with respect to the graph norm

Let $T:A\subset E\to F$ be a linear closed operator with dense domain. (E,F are Banach spaces) and $x_n$ a sequence in A such that $||x_n-x||_E\to 0$. Let $||x||_1:=||x||_E+||T(x)||_F$ be the graph ...
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2answers
31 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
2
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1answer
35 views

Definition of the limit of a sequence

I'm looking over the following definition of convergent limits: A sequence $(x_n)$ in $\mathbb{R}$ is said to converge to $x \in \mathbb{R}$, or x is said to be a limit of $(x_n)$, if for every ...
0
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2answers
33 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
0
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1answer
36 views

Test whether $\sum_{n=1}^{\infty}\frac{\ln{n}}{n}$ converges or diverges

I am trying to solve this using an integral test, but I am unsure whether or not this is correct. Let $f:[2,\infty)\to\mathbb{R}$ be defined by $f(t)=\frac{\ln{(t)}}{t} >0\ \forall t\geq2$. Now ...
0
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0answers
27 views

Uniform convergence on subintervals

(a) Fix a positive integer $M$ and let $\{f_{n} : [0, M]\rightarrow \mathbb{R}\}$ be a sequence of functions. Suppose that $f_{n}\rightarrow f$ pointwise on $[0, M]$ and that $f_{n}\rightarrow f$ ...
0
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2answers
21 views

Determine the interval of convergence of series $\sum_{n=1}^{\infty } \frac{1}{\cos ^2(n \cdot x)+\sqrt{n}}$

So, hey. I was sincerely trying to find it by myself with Weierstrass M-test, but failed occasionally, because I ended up with $\sum_{n=1}^{\infty } \frac{1}{\sqrt{n}}$,which is a divergent series. ...
2
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1answer
24 views

Convergence of $\sum(-1)^k\frac{(\ln k)^p}{k^q}$ where $p,q$ in positive $\mathbb{R}$

For any $p, q$ in positive $\mathbb{R}$ $$\sum_{k=2}^{\infty}(-1)^k\frac{(\ln k)^p}{k^q}$$ I want to Use alternative series test for convergence but I'm struggling to verify that $\frac{(\ln ...
1
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1answer
25 views

Find sum of this convergent series

find the sum of the infinite series $\sum_{i=0}^\infty\frac{2^i}{n^{(2^i)}}$ for $n>1$ I tried the following $\frac{1}{n}+\frac{2}{n^2}+\frac{4}{n^4}+...=k$ ...
0
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3answers
40 views

How can I determine if the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges or diverges?

Determine whether the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges. If it converges find the limit and if it diverges determine whether it has an infinite limit. Proof: let $a_{n} = ...
3
votes
1answer
34 views

Implications of some sort of $l^2$/uniform convergence

Sorry about the title, but I couldn't really figure out how to describe my problem in one sentence... I'm having some problems with real limits: For $f,g : \mathbb{N} \to \mathbb{R}$ let ...
1
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0answers
26 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
2
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1answer
37 views

Proving the convergence of a recursive sequence

Consider the following sequence, defined recursively: $$ x_{n+1}=\frac{2x_n^3+2}{3x_n^2} $$ Prove that $x_n$ converges to $ 2^{1/3} $ and $ x_7 $ approximates $ \root 3 \of2 $ accurately to 6 ...
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0answers
23 views

White noise, how is its definition sensical

White noise is defined as as noise containing all frequencies. Now, consider the inverse fourier transform of white noise, $R$ being the fourier transoform of the noise: $$\int_{-\infty}^\infty R ...
0
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1answer
23 views

Convergence of $\sum \frac{i^n}{\log(n)}$

Study the convergence of $$\sum_{n \geq 2 } \frac{i^n}{\log(n)} $$ where $\log$ of course denotes the 'natural logarithm' and $i \in \mathbb{C}$ Oddly enough I managed to show Abel's Test for ...
0
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0answers
26 views

Proving the convergence of this sequence

So I've asked myself the question "If a sequence converges, does the series of distances between the consecutive elements converge?". As a countexample I came up with the idea of a sequence ...
3
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1answer
48 views

Convergence of $\sum x_k \log (x_k^{-1}) $ [on hold]

Prove or disprove: If $\{x_k\} \subset (0,1)$, and $\sum x_k < \infty$, then $\sum x_k \log (x_k^{-1}) < \infty$. What if $\{x_k\}$ is monotone? Thanks a lot.
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1answer
20 views

Does $E(|X_n - X|) \rightarrow 0$ implies $X_n$ converges in probability to $X$?

I think it does, I've tried proving it by using Chebishev's Inequality but it only prove that it works with quadratic convergence and I can't adapt it... Can you help me please? Thank you very much! ...
4
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2answers
95 views

Find the limit $\,\,\, \lim_{n \to \infty}\Big(\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\Big)^{1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
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3answers
40 views

Is sequence limited and what is limit

I am stuck at one problem. So I have to check if sequence is convergent. $$\frac{2^x}{x!}$$ My thinking was to calculate limit and if limit exists it's convergent, but I am struggling with this: ...
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0answers
8 views

Speed of pseudo-inverse (with possibly ill-conditioned matrices)

I am computing the pseudo-inverse of several matrices of identical size $m \times n$ . However, computation (e.g. with the LAPACK pinv) seems to be much slower in some cases (5 to 10 times slower). ...
4
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3answers
34 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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2answers
42 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
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0answers
12 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
0
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2answers
40 views

How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$?

In Baby Rudin, Theorem 3.27 on page 61 reads the following: Suppose $a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. Then the seires $\sum_{n=1}^\infty a_n$ converges if and only if the series $$ ...
2
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1answer
29 views

How should Theorem 3.22 in Baby Rudin be modified so as to yield Theorem 3.23 as a special case?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition, and am at Theorem 3.22. Theorem 3.22: Let $\{a_n\}$ be a sequence of complex numbers. Then $\sum a_n$ converges if and ...
1
vote
1answer
18 views

convergence criteria of an infinite series

$\sum _{n=1}^{\infty }{\frac {1}{50}}\,{\frac { \left( -1 \right) ^{1+n }{\it a}\, \left( 10000\,\cos \left( tn \right) \epsilon\,\delta_{{ 1}}-10000\,\cos \left( \frac{1}{10}\,\sqrt {4201}t \right) ...
3
votes
2answers
72 views

Is the following series convergent $\sum_{n=1}^{\infty}\dfrac{2^n+3^n}{3^n+4^n}$

Is the following series convergent $\sum_{n=1}^{\infty}\dfrac{2^n+3^n}{3^n+4^n}$ I tried everything, nothing appears to work. can some one give an idea
2
votes
1answer
21 views

Series of Sequence which always diverges

Suppose {$a_n$} is a sequence with $a_n>0$. For each $k$ in $\Bbb{N}$, set $$b_k = \frac{1}{k} \sum_{n=1}^{k}a_n$$ then woud $\sum_{k=1}^{\infty}b_k$ always diverge? I want to use Converge ...
2
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0answers
41 views

Convergence of $\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$ [duplicate]

Is following sum convergent? $$\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$$ Integral test, Dirichlet test doesn't apply. Any idea !
0
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1answer
12 views

a bounded function is converge in measure, then its limit is also converge

If a series function, ${f_n} \rightarrow f $ in measure $\mu$, and $|f_n| \leq M$, how to show that $|f| \leq M$? My instructor gave a hint as follows, but I do not believe the first inequality ...