Convergence of sequences and different modes of convergence.

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Series Proof Question [duplicate]

By considering the partial sums for S, that is Sn =1+2+3+···n show that the infinite series S does not converge. However in this video http://www.numberphile.com/videos/analytical_continuation1.html ...
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2answers
65 views

Do the following series converge if $a_n>0$ and $ \sum_{n=1}^{\infty}a_n$ diverges?

Do the following series converge if $a_n>0$ and $\sum_{n=1}^{\infty}a_n$ diverges ? a.) $\sum_{n=1}^{\infty}{a_n \over 1+ a_n}$ b.)$\sum_{n=1}^{\infty}{a_n\over 1+ a_n ^2}$ ...
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4answers
42 views

If $x_n \to a$ and $x'_n \to a$, then $\{x_1, x'_1, x_2, x'_2, …\} \to a$

I know that if all subsequences of $\{x_1, x'_1, x_2, x'_2, ...\}$ converge to $a$, then $\{x_1, x'_1, x_2, x'_2, ...\}$ converges to $a$, but I only know two subsequences of $\{x_1, x'_1, x_2, x'_2, ...
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1answer
25 views

Uniform continuity preserves uniqueness of convergent sequences?

If $f: (a, b) \to \Bbb R$ is uniformly continuous, $\{x_n\}$ and $\{x'_n\}$ are sequences in $(a, b)$ with $x_n \to b$, $x'_n \to b$, $f(x_n) \to y$, and $f(x'_n) \to \overline{y}$, prove $y = y'$. ...
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1answer
39 views

How to show $ \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}})$ converges?

Let $a_{n} \ge 0 \hspace{1cm} \forall n \in$ $ \mathbb{N} \cup \{0\}$. and $ \sum_{n=1}^{\infty} a_{n}$ converges and $ b_{n}=\sum_{k=n}^{\infty} a_{k} $ Then we have to prove that$ ...
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1answer
35 views

Convergence of a sequence 4

Suppose there is a sequence $\{ x_n \}$. Let us define another sequence $\{ y_n \} $ such that $$y_n=2x_{n+1} - x_n$$ Please prove that if $\{y_n\}$ converges to $L$, then $\{x_n\}$ is convergent and ...
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1answer
56 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
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2answers
103 views

Integral test for convergence proof

Can someone help me understand this proof? I don't understand why $f(n+1) = \int_n^{n+1}{f(n+1)}$ Thank you so much and I am sorry I have nothing else to contribute as I'm fearing it is a ...
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1answer
29 views

Find a sequence $a_n$ such that its sum converges, but the sum of its logarithm doesn't - Generalisation Help

This question came up in a past paper that I was doing, but it seems to be a fairly common, standard question. Give an example of a sequence $a_n$ such that $\sum(a_n)$ converges, but ...
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33 views

Expected value problem… Infinite series with factorial

In an exercise I have to calculate the following expected value: $$ \sum_{i=4}^\infty500\frac{e^{-4}4^i}{i!} $$ Leaving $500e^{-4}$ outside and using D'Alambert's criterion (with $a_n = ...
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40 views

Unifrom Convergence of series of product of two sequences

Suppose {$f_n$}, {$g_n$} defined on $E$ and, (a) $\Sigma f_n$ has uniformly bounded partial sums; (b) $g_n \to 0$ uniformly on $E$ (c) $g_1(x)\geq g_2(x)\geq g_3(x)\geq ...$ for every $x \in E$. ...
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2answers
109 views

Convergence test for a special type of series

I have two null sequences $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ with $a_n \ge 0$ and $b_n \le 0$ for all $n\in \mathbb N$. Let $(c_n)_{n\in\mathbb N}$ be a sequence with either $c_n=a_n$ ...
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1answer
44 views

Interesting Power Series

The series is $\sum_{n=1}^{\infty} r(n)x^n$ , where $r(n)$ is defined as the divisor function. The question is , what is the radius of convergence of the power series? Maybe it is not that interesting ...
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1answer
34 views

What is the difference for the convergence? What is correct?

Suppose we have a Lebesgue integrable function, $f\in L^1(\mathbb{R}^d)$. I would like to approximate it by nice functions for instance $\{f_n\}_{n\geq 1}\subset C_0^\infty(\mathbb{R}^d)$ smooth ...
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1answer
44 views

Convergence or divergence of $\int_0^a \frac{\cos \frac{1}{x}}{x \sqrt{x}}\mathrm dx$

I want to check if the following integral exists: $$\int_0^a \frac{\cos \frac{1}{x}}{x \sqrt{x}}\mathrm dx$$ I tried to bound it from above by a converging integral with no success. Any hint?
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3answers
64 views

Prove $(s_n)$ converges, where $s_{n+1} \lt r(s_n)$.

Suppose $(s_n)$ is a sequence such that $s_{n+1} < r s_n$ where $r$ is a constant with $0<r<1$. Prove that $s_n$ converges. All values are positive. I thought about maybe solving this ...
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2answers
67 views

Showing $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$ is Cauchy.

Let $(s_n)$ be a sequence defined as $s_1 = 1, s_2 = 2$ , and $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$. Prove that $(s_n)$ is Cauchy. I can see how it is convergent and Cauchy but not sure ...
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1answer
31 views

Convergence radius of complex power series

If $a_n\neq 0$ for all $n \geq n_0$ and $\lim|\frac{b_n}{a_n}|=1$, then $\rho(S)=\rho(T)$. Since S=$\sum a_nz^n$ and T=$\sum b_nz^n$. I tried to use the definition of convergence radius $$\limsup ...
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Series convergence question

This question occured to me in the context of ARMA time series analysis: Let $\alpha_n\geq 0,\, n= 1,2,\dots$ such that $\sum_{n=1}^\infty\alpha_n = 1$ and define the sequence ...
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52 views

Limit of a sequence of Marlov processes

Let $(X_n)$ be a sequence of Markov processes on, say, $[0,1]$, that converges in finite dimensional distributions to a process $X$. Is it true that $X$ must also be a Markov process ?
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$\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.

Suppose that the series $\sum_{n=1}^{\infty}a_{n}$ converges while $\sum_{n=1}^{\infty}a_{n}^2$ diverges. Prove that $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.
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34 views

Convergence of $\int^{+\infty}_{0} \frac{\arctan x }{x^a} \sin x\, dx$

I´m trying to find out if this integral is convergent (and for what values of $a$) or not: \begin{equation*} \int^{+\infty}_0 \frac{\arctan x}{x^a} \sin x \ dx,\quad a \in \mathbb R \end{equation*} ...
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2answers
33 views

Uniform convergence of sequence of functions with infinite roots to a limit with finite roots

Consider a sequence of continuous functions $(f_n)$ defined over $[0,1]$ such that, for all $n$, the set: $$A_n = \{x\in [0,1] : f_n(x) = 0\}$$ is infinite in cardinality. Can $(f_n)$ uniformly ...
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1answer
17 views

Determing taylor series from other series

Consider $\cos(x)$ and $\cos(3x^2)$. How to determine the latter's Taylor series from the formers at $a = 0$? I'd write $$\cos{x} = \sum_0^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$ Now, I could just ...
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1answer
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On convergence in probability given a bound on the random variable.

I am dealing with the end of a proof: Could somebody please clarify for me the extra steps needed to show that $P(||\bar{W}_n - \mu||_{\infty} > 3\varepsilon) \rightarrow 0$ as $n \rightarrow ...
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Convergence in the Tychonoff topology on $\mathbb{R}^\mathbb{R}$.

This is Example 10.2(b) (p.70) in Willard's General Topology: In the product space $\mathbb{R}^{\mathbb{R}}$, a sequence $f_n$ converges to $f$ iff $f_n(x) \rightarrow f(x)$ for each $x \in ...
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266 views

Fermat's last theorem generalization [closed]

Conjecture: Let $g$ is a positive algebraic number greater than two, then the equation ($x^g+y^g=z^g$) doesn't have any solution, where ($x, y$ and $z$) are three distinct positive coprime integers ...
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179 views

Convergence of Ornstein-Uhlenbeck process as a scaled Brownian Motion

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
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3answers
61 views

Does this integral converge or diverge?

I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence ...
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30 views

Help with when the integral is convergent

quick question, how can you tell for which a the integral $$\int_1^{\pi/2} \dfrac{\cos^2(2x) - e^{-4x^2}}{x^a\tan{x}}dx$$ is convergent? Thanks in advance.
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Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
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36 views

A special limit-sum interversion

Let $l\in\mathbb{R}$ and $(u_k)_{k\in\mathbb{N}}$ a sequence converging to $l$. Let $a_{n,k\in\mathbb{N}}$ be such that $\displaystyle\forall n\in\mathbb{N},\sum_{k=1}^na_{n,k}=1$. $\forall ...
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prove $\{z_n\}$ converges where $z_n=f(z_{n-1})$

Let $f : \mathbb C \rightarrow \mathbb C$, $f(z) = \frac{1}{2}z^2 + 1$, and $c = 1 + i$. Let $\{z_n\}$ be the sequence defined by iterating $f$ on some initial value $z_0 ∈ C$ (that is, $z_n = ...
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for which alpha is the Integral convergence

Let $\alpha>0$ and $$ f(x)=\frac{\ln x}{(x-1)^{\alpha}} $$ for $x>1$ i found that for $\int_2^{\infty}f(x) dx$ the integral is convergence for $\alpha > 2$ but for which $\alpha$ is ...
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3answers
243 views

Prove if the following sequence is convergent

$$ f(n) =\frac{6n^3 + 2n+(−4)^n}{4^n-1} $$ The sequence is dominated by $4^n$, so we divide by the dominant term, and we get $(-1)^n$ which is not a null sequence. So it is not monotonic and not ...
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Proving that the mean of a random variable is continuous, where is dominated convergence being used?

I am looking at the proof of the first part of this lemma. Previously in the text another theorem was stated: Convergence in distribution, $Y_n \implies Y$, holds iff $Ef(Y_n) \rightarrow Ef(Y)$ ...
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Show by comparison that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ [duplicate]

So, I have to show (as the title says), that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ by comparison. I've managed to show it for ...
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For which $\alpha$ does $\int_2^{\infty} \dfrac{\ln(x)}{(x-1)^{\alpha}} dx$ converge?

So, as the title says, I have to show which $\alpha$ makes $\int_2^{\infty} \dfrac{\ln(x)}{(x-1)^{\alpha}} dx$ converge? I have really have no idea how to do this. I've managed to show that it is ...
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1answer
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Prove that the sequence is Cauchy

This is problem 17, page 55, from section 1.2: Cauchy Sequences in the textbook Introduction to Analysis, Fifth Edition, by Edward D. Gaughan. Prove that the sequence ...
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29 views

Analysis: Basic Sequence Proof

Prove that, if $\left\{a_n\right\}_{n=1}^{\infty}$ converges to A, then $\left\{|a_n|\right\}_{n=1}^{\infty}$ converges to |A|. Is the converse true?
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Almost sure convergence and L1 convergence

I am preparing myself for the mid-term exam of my probability theory exam, and am solving questions from previous years exams. One of these questions I couldn't answer, and so far I haven't found ...
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75 views

The convergence of $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$ with two different tests.

Let $p,q\in\mathbb{R}$ and consider the series $\sum_{n\geq 2}\frac{1}{n^{p}(\ln(n))^{q}}$. i) Show by the comparison test, that the series is convergent if $p>1$ and divergent if ...
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1answer
42 views

Limits and Convergence

Prove: If $\lim \limits_{n \to \infty} a_{n+1}-a_n = 0$ then $a_n$ has to converge. I understand that the distance between adjacent $a_n$ elements approaches $0$. Since $a_{n+1}-a_n$ converges it has ...
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Check whether function series is convergent

I have the following task: Check whether given function series is pointwise convergent, uniformly convergent or "almost" uniformly convergent (id est $f, f_{n} : I \rightarrow \mathbb{R} $ and $ ...
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5answers
37 views

Calculus 2 Series Convergence - Can I Use Comparison Test?

Can I use the comparison test for the following problem? $$\sum _{n=1}^{\infty }\:\frac{1}{n^2-6n+10}$$ The denominator has a negative coefficient so i'm not sure if its valid to compare it to a ...
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1answer
46 views

Prove that $f_n(x)=\frac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$

Title says it all; I have to prove that the function sequence $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$, with ...
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0answers
19 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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1answer
55 views

Convergence of $\sum\limits_{n=1}^{\infty}\left (1-n\sin \frac{1}{n}\right)^\alpha$ for parameter $\alpha$

For exactly which real values of $\alpha$ is the series $$\sum_{n=1}^{\infty}\left(1-n\sin \frac{1}{n}\right)^\alpha$$ convergent? Please give some hints.
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1answer
40 views

How is the convergent sequence $\frac{1}{n-1}$ bounded?

In a metric space all convergent sequences are bounded. This example in the real numbers should then be bounded but, it is infinite at n=1 so I do not understand how this can be true. In the proof ...
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1answer
30 views

Show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b,$ and $x \in [a,b]$ is not uniformly convergent

So, I have to show that $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ with $0<a<b$ and $x \in [a,b]$ is pointwise convergent, but not uniformly convergent. The pointwise convergence is pretty straight ...