Convergence of sequences and different modes of convergence.

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4
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2answers
80 views

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0.

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0. I'm missing something. I doubt that I'm really doing anything from " $\frac{1}{n+1}$ " on. I set it $< ...
2
votes
1answer
32 views

Convergence of alternating series not subject to alternating series test

Given two $a_n,\; b_n>0$ such that $lim_{n \to \infty} a_n,\; b_n=0$ and $\lim_{n \to \infty} \frac{a_n}{b_n}=1$, where neither series is necessarily monotonic: if $\displaystyle \sum_{k=1}^\infty ...
2
votes
5answers
109 views

Determining convergence or divergence of series

I am wondering the convergence or divergence following series $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{ n+\sin (n)} \\ $$ My 1st attempt is 'alternating series test' $$ $$ But, ...
2
votes
1answer
33 views

Proving that $\int_0^t s(x) dx = \frac{t^2}{1-t}$

Let $ s =s(x)$ given by $s(x) = \sum_{k=1}^\infty (k+1)x^k. $ Prove that for all $ t \in ]-1,1[,$ $$\int_0^t s(x) dx = \frac{t^2}{1-t}$$ Conclude that, for all $x \in ]-1,1[,$ $$\sum_{k=1}^\infty ...
3
votes
1answer
36 views

Does 'bounded by convergent series' imply convergence?

Suppose we have a real series $\sum x_{n}$ which is convergent. If we have either $0<y_{n}<x_{n}$ or $x_{n}<y_{n}<0$ for all $n$ past some limit (so $|y_{n}|<|x_{n}|$, and they have ...
0
votes
1answer
31 views

Determining Rate of Convergence

I have a question from the homework here: Show that the following sequence converges linearly to 0 $$P_n = \frac{1}{n^2}; n \ge 1$$ So we know $$\lim\limits_{x \to \inf} \frac{|p_{n+1} - ...
2
votes
2answers
53 views

A convergence test similar to Gauss' test.

Consider a sequence of complex numbers $(a_n)$ and assume that we can write $$\frac{a_{n+1}}{a_n}=1+\frac{\lambda}n+\frac{b_n}{n^2}$$ where $b_n$ is bounded and $\Re(\lambda)<-1$. Can we show that ...
0
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1answer
53 views

Convergence of $\sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2}$

Does the series $$ \sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2} $$ converge? The ratio test is inconclusive, so I think I must use the comparison test. But I couldn't ...
7
votes
0answers
65 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
1
vote
2answers
57 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
4
votes
1answer
107 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
2
votes
2answers
28 views

Suppose that $f(x) \ge 0$ and $\lim_{x \to c} f(x) = L$. Prove $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$

Suppose that $f(x) \ge 0$ in some deleted neighborhood of $c$, and that $\lim_{x \to c} f(x) = L$. Prove that $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$ under the two different assumptions on $L$: $L=0$ ...
1
vote
1answer
15 views

Convergence of an Iterative Sequence…

Let $g(x)=\frac{2+x}{1+x}$. Set now the sequence $(x_n)_{n\in\Bbb{N}}$ such that $x_0=0$ and $x_{n+1}=g(x_n)$. Show that this sequence converges and, furthermore, converges to $\sqrt[]{2}$.
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0answers
18 views

Convergence of a Double Sum over 2 integers

Does the following double summation over x, x' (both integers) converge? $\sum\limits_{x=-\infty}^\infty \sum\limits_{x'=-\infty}^\infty \frac{Sin^2(2 \pi(x-x'))}{(x-x')^2}$. If so evaluate the sum. ...
0
votes
1answer
28 views

Cauchy sum of ratio of sequences not converging.

Prove if ${x_n}$ and ${y_n}$ are Cauchy and $x_n + y_n > 0$, for all natural n, then $\frac{1}{x_n + y_n}$ cannot converge to zero. Attempt: Suppose $x_n → a$ and $y_n → b$. Then $x_n + y_n → a + ...
1
vote
1answer
18 views

Bounded intervals, sequentially compact.

Can someone please give me an example of a bounded interval in R that are not sequentially compact? A subset E of R is said to be sequentially compact if and only if every sequence x_n in E has a ...
3
votes
1answer
53 views

Proving convergence of $\sum \frac{\sin n}{2^n}$

Prove that the following sequence ($x_n$) is convergent: $$ x_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} + ... + \frac{\sin n}{2^n} $$ I have tried to use to the sequence is ...
0
votes
1answer
20 views

Radius of Convergence (Non-Series)

I am confronted with the following exercise: Compute the radius of convergence for the expansion at the point $z=4+4i$ for \begin{equation} f(z)=\frac{z^{5}e^{z}}{(2-z)(3i-z)} \end{equation} I ...
0
votes
0answers
10 views

Convergence of normalized stochastic integral

I am wondering about some results about the convergence of processes like that : $$ \frac{1}{T} \int_{0}^{T} H_{s}dM_{s} $$ with $M_{s}$ a semi-martingale when T goes to $ +\infty$ Thanks a lot :-) ...
1
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1answer
27 views

Converse of alternating series test

Is the converse of the alternating series test true? In other words, given a sequence $a_n>0$, with neither $a_{2n}$ nor $a_{2n-1}$ constant, for which there exists no positive $N$ such that ...
0
votes
1answer
27 views

Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
0
votes
0answers
24 views

Convergence of probability density in the tail.

Let $f(x)$ be a probability density with respect to the Lebesgue measure. The distribution has first moment, e.g. $\int_{-\infty}^\infty |x| f(x) dx < \infty$. Further assume that there exists $K$ ...
0
votes
2answers
36 views

Convergence of square of monotone sequence implies convergence of sequence.

Suppose $(a_n)$ is a monotone sequence. Prove that if $(a_n^2)$ is convergent, then so is $(a_n)$. How do I use the monotone convergence theorem to prove this?
6
votes
2answers
131 views

How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?

I'm trying to show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m)) + \cdots$ converges for all real numbers $m$. To be specific, the series is defined as follows: $\sum_1^\infty{a_k}$ ...
1
vote
2answers
26 views

$f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
0
votes
1answer
29 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...
1
vote
3answers
78 views

Divergence of $n!^{1/n}$ [duplicate]

I'm trying to find a simple proof that $n!^{1/n}$ is a divergent sequence. I have proved it using a lower bound you can get from an integral (or Stirling's approximation) that $n!^{1/n}>\ln(n)$, ...
1
vote
1answer
56 views

Prove that x{n} is convergent

So I'm currently studying for my midterms and I found the following question from my practice set which I'm unable to solve: Prove that the following sequence $(x_{n})$ is convergent. Let $$x_{n} = ...
0
votes
2answers
55 views

Prove that limit exists and is finite, sum of series, Cauchy.

Prove that $\lim_{n \to \infty} \sum_{k=1}^n \frac{(-1)^k}{k}$ exists and is finite. Attempt: Suppose $\{x_n\}$ is real sequence, and $x_n = \frac{(-1)^k}{k}$. I know if I prove that it is Cauchy, ...
0
votes
1answer
32 views

Cauchy sequence and metrics

I'm having trouble with another analysis homework problem: Let $x_n$ be a sequence in $\mathbb{R}$ such that $d(x_n, x_{n+1}) \le \frac{d(x_{n-1},x_n)}{2}$. Show that $x_n$ is a Cauchy sequence. I ...
2
votes
1answer
35 views

Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
3
votes
1answer
40 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
2
votes
1answer
47 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
0
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0answers
3 views

how to define the degree of aggregation/convergence for a bunch of two-dimensional points

Every point has an uncertain position with error. If their positions have large dispersion, they are considered to be multiple points. Otherwise, they can be considered to be a single point. In ...
0
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2answers
50 views

The integral $\int_0^\infty f(x)g(x) e^{-x}\,dx$ is convergent for all real polynomials $f,g$ [duplicate]

For all real polynomials $f$, $g$ Why is the integral $\int_0^\infty f(x)g(x) e^{-x}\,dx$ convergent?
0
votes
1answer
37 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
2
votes
2answers
71 views

How can I prove that $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent?

I'm trying to prove that the sum $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent. I've tried Cauchy's root test - but I get the limit to be 1, so the test is inconclusive. I also ...
0
votes
2answers
17 views

Radius convergence of a power series…

"Suppose that $\sum_{n=0}^{+\infty}a_nx^n$ has convergence radius $R$, $R>0, \text{or }R=+\infty$. Proof that the convergence radius of $\sum_{n=0}^{+\infty}na_nx^{n-1}$ is also $R$." This seems ...
5
votes
2answers
135 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, ...
0
votes
1answer
21 views

Almost Sure Convergence for Sample Mean of Bernoullis

Let {$B_i$} be a sequence of Bernoulli($\mu$) variables and $X_n$ its sample mean $X_n=\frac1n\sum_i^nBi$. Because of the Strong Law of Large Numbers, we know that $X_n$ converges almost surely to ...
0
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2answers
24 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
1
vote
1answer
20 views

Finding the convergence radius of a power series

Let $\sum\limits_{n=0}^{+\infty}a_nx^n$ be a power series. Prove that If $\large \lim\limits_{n\to\infty}a_ns^n=0,s>0$, then the power series above converges absolutely for $|x|<s$. ...
0
votes
1answer
44 views

Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
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vote
2answers
63 views

Does multiplication preserve almost sure convergence?

Suppose $(\xi_i)$ is an iid sequence and $(V_i)$ is a sequence such that $V_n$ converges almost surely to zero. Then $\xi_nV_n$ converges almost surely to zero or in probability? How can I prove this? ...
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vote
3answers
51 views

Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge

This is a homework problem for an undergrad topology course. Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in ...
2
votes
1answer
35 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
-1
votes
1answer
31 views

Proving that a sequence is Cauchy on the basis of squeeze theorem

Let $\{x_n\}$ be a sequence of real numbers such that $$|x_n| \leq \frac{2n^2 + 3}{n^3 + 5n^2 +3n + 1}$$ Prove $\{x_n\}$ is a Cauchy sequence Proof: Suppose that ${x_n}$ be a sequence of real ...
2
votes
1answer
48 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
1
vote
2answers
44 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
1
vote
1answer
45 views

Is there a contradiction in these two definitions of limit Superior?

Definition 1 : Let $A=\{a_n\}$ be a sequence of real numbers not necessarily bounded . Then we define : $\lim \sup ~ a_n = \inf ~\{\sup ~a_n,~\sup ~a_{n+1}~\sup ~a_{n+2}, \cdots\} $ Definition 2 : ...