Convergence of sequences and different modes of convergence.

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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 ...
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43 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{n→\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
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30 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 ...
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18 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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48 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 $\lim_{x\rightarrow \infty }e^x/p(x)=1$?
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121 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 ...
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9 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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30 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 ...
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34 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 ...
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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: ...
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33 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 ...
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26 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 $f_{n}\rightarrow f$ ...
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20 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. ...
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22 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 ...
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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 $\frac{1}{n}+\frac{2}{n^2}+\frac{4}{n^4}+...=k$ ...
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39 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} = ...
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31 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 ...
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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 ...
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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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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 ...
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21 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 ...
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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 ...
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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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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! ...
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92 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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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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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). ...
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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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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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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, ...
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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 $$ ...
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28 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 ...
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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) ...
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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
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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 ...
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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 !
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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 ...
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22 views

Do convergence in distribution along with uniform integrability imply convergence of mean?

Thorem about necessary and sufficient condition for $L_1$ convergence states that $X_n$ - non negative, $EX_n$ converges to $EX$ if and only if $X_n$ converges to $X$ in probability and $X_n$ is ...
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13 views

Convergence in distribution example

I have a trouble understanding how below is true: I have drawn $F_{\frac{Y_n}{n}} (y)$ above. Is this correct? Now, if I send n to infinity, I still get the same graph (discrete) and not the ...
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1answer
20 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
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1answer
43 views

A proof on uniform convergence of polynomials with bounded degree

In $[0,1]$ suppose that you have a sequence of polynomials $(P_n)_{n\in\mathbb{N}}$ of at most degree $M$ each. Also, suppose that $P_n(x) \rightarrow 0$ pointwise for every $x\in[0,1]$. Is is true ...
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85 views

Proving a strange identity

Numerically, it would seem the following identity holds true: $$\frac{6}{7}=\lim_{n\to\infty}\sqrt[n]{\sum_{k=3}^\infty{\left(k-\sum_{j=1}^{k}\frac{1}{j}\right)^{-n}}}$$ Down below I have proven ...
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28 views

How to state convergence through limit comparison test?

I am able to show convergence of the following series through the root test but am trying to practice the limit comparison test and can't figure out how to do it that way. $$\sum_{n=1}^\infty ...
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89 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
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1answer
23 views

Determining values of k for which a series is absolute convergent/conditionally convergent/divergent

So I have a series $\sum_{n=k}^\infty \frac {(-1)^n}{n \choose k}$ = $\sum_{n=k}^\infty \frac{(-1)^nk!(n-k)!}{n!}$ and I want to know for which values of k the series converges absolutely, for which k ...
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28 views

Conditions of Convergence of a Sequence

$$d_{n}=\sqrt[n]{A^n+B}$$ If $d_{n}$ converges, find an expression for its limit. Explain your conditions of convergence in terms of both numbers and their relationship with each other e.g. "If $A ...
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54 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
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1answer
22 views

True or False, sequences converging.

True or False? For any positive real number r, {r^n} converges. False: Take any positive r, then as n → ∞ r diverges. If {x_n*y_n} converges, then {x_n} and {y_n} both converge. False: Suppose ...
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1answer
13 views

Proving a limit converges to an equation

Verify that $\lim{\epsilon→0}$ of the given equation converges to the equation of $y=xe^{px}$ $$y=\frac{-1}{\epsilon}e^{px}+\frac{1}{\epsilon}e^{(p+\epsilon)x}$$ After I finish my work, I ended up ...
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1answer
24 views

Comparing plots question

I have a program that randomly generates line plots and what I would like to do now is compare two of those line plots and get a measure of 'similarity' between them. Now I feel as if this measure of ...