Convergence of sequences and different modes of convergence.

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Convergence of series $\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} )}}{(n+\frac{1}{n} )^{\frac{1}{n} }}$

i need help for find method or methods for solve this series and find the convergence. I very appreciate for any help and yours comments. $$\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} ...
0
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1answer
17 views

Definite integral convergence answer check

this problem is part of my homework assignment and I would like you guys to see if I'm right. So I need to find out for which $a$ is the integral ...
0
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2answers
37 views

Convergence of $\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$

I can't analyse the convergence of this integral: $$\int_{2}^{+\infty} \frac1{x \ln^\alpha x}dx$$ with $\alpha \in R$. I have tried to find some functions and use comparison theorem, but I haven't ...
3
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2answers
59 views

How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$

I'm trying to prove this sequence converges: $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ I noticed that this is continuous function which its derivative is always less than $0$ for $ x \gt 1 $, so I ...
0
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1answer
32 views

Super-Linear Growth after Reordering

Suppose we have a sequence/function $f: \mathbb{N} \to \mathbb{N}$. (For the scope of this question, let $\mathbb{N} = \{ 0, 1, 2, \dots \}$.) From $f$, we are going to create a function $g: ...
0
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2answers
34 views

Convergence of random variable times function: $nX_n$

If $X_n\xrightarrow[]{p}X$, can I prove that $n(X_n-X)\xrightarrow[]{p}0$ if $X$ is a natural number. I know that if $Y_n$ is bounded in probability $Y_nX_n\xrightarrow[]{p}0$, or that if $n$ is a ...
0
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0answers
15 views

Composition of Lp convergent function and continuous function

Let $f$ be a continuous function on $\mathbb R$ such that for $U\subset \mathbb R ^n$ bounded it holds that $\forall w\in L^p(U) ~~ f(w)\in L^q(U)$. Let $~u_k \rightarrow u$ in $L^p(U)$ . Does ...
1
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0answers
41 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
0
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3answers
69 views

Do their exist power series with non circular regions of convergence?

So far just about any series of the form $$ \sum_{i=0}^{\infty} \left(a_ix^i \right)$$ Has tended to have a circular disk of convergence (of some radius, sometimes even 0). Is there a reason this ...
0
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2answers
39 views

Fixed points that are NOT convergent points

Are there any fixed points that are NOT converget (aka attractig fixed points) in the sequence $x_n = 5\ln x_{n-1}$? How do you determine this?
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0answers
70 views

$l_1$ space not being inner product space

Hello so I am just reading some book in functional analysis but there is some calculations that I am getting different results for, so in the book he tries to prove that indeed $l_1$ is normed space ...
2
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2answers
219 views

When is this integral convergent?

Let $a \in \mathbb{C}$. Consider the integral $$\int_{-\infty}^{+\infty} \frac{e^{-ax}}{1 + e^x} dx,$$ for which values of $a$ is this convergent? Is it right to say that $a$ has to be purely ...
0
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0answers
51 views

Almost sure convergence, product of variables

Consider the sequence of random variable given by distribution : $$\mathbb{P}(X_n=1)=1-\frac{1}{n},$$ $$\mathbb{P}(X_{n}=0)=\frac{1}{n}$$ and $Y_n=X_n \cdot Y$ for random variable Y. Does the $Y_{n}$ ...
1
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0answers
23 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
0
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1answer
51 views

Almost sure convergence of random variables

$(X_n)$ is a sequence of random variables having the following distribution: $$P(X_n=1)=1- \frac{1}{n},\; P(X_n=0)=\frac{1}{n}$$ (we don't assume that those variables are independent). $X$ is some ...
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1answer
36 views

Convergence of conditional probability

Can anyone help me with this question: suppose $X$ and $Y$ are non-negative random variables. Under what condition does $\lim_{\delta\rightarrow 0}\dfrac{P(Y\leq ...
0
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1answer
53 views

Convergance of some series of random variables

Random variables $(X_{n})$ are independent and have the distribution $P(X_{n}=1)=p, P(X_n=-1)=1-p$, $\frac{1}{2}<p<1$. Prove that $$X_1+X_2+\dots+X_n \to \infty $$ almost sure. Let ...
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1answer
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
2
votes
2answers
189 views

Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$

Prove that the sequence $s_n = \frac{1}{n}$ converges to $0$. I am writing this proof in order to help other people to understand better how to prove if a sequence converges and in particular why ...
3
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2answers
55 views

Show that $\frac{X_1+\dots+X_n}{n}$ converges to $\infty$ a.s. for $X_n \sim U([0,n])$ independent

Random variables $(X_{n})$ are independent and $X_{n}$ has an uniform distribution on $[0,n]$ for n=1,2,... Prove that: $$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}\rightarrow \infty$$ almost sure. We can ...
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2answers
26 views

Find the domain of convergence of series $\sum_{n=1}^{\infty }(nx)^n$

Find the domain of convergence of series $$\sum_{n=1}^{\infty }(nx)^n$$ I tried to find it by using the Root Test as follows: $$=(nx)^{n/n}=nx$$ I know for convergency, the $(nx)$ should be less ...
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1answer
92 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{5^n + 2^n}{n!})$ converges using perhaps the D'Alembert test, but it doesn't really seem to fit..are there other ways? ...
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1answer
58 views

$\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges using perhaps the D'Alembert test, but given the sine I cant really see it happening..are there ...
1
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1answer
27 views

Exponential(1) distributed random variable convergence

I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random variables. Show that $n^\alpha ...
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5answers
82 views

Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent

Show if $\sum_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt{n}}$ is absolute convergent or divergent First i subbed numbers in $$\lim_{n \to \infty} \frac{(-1)^n}{1+\sqrt{n}} = \frac{-1}{1+\sqrt{1}} + ...
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1answer
24 views

Does this contravene the dominated convergence theorem?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. $f(x):=\lim\limits_{n \rightarrow ...
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2answers
43 views

Analysis: limit of $x^n$ if $x\gt 1$ [closed]

could someone tell me how I can show $x^n \to \infty$ if $x \gt 1$ From limit to infinity definition $x^n \to \infty$ if for all $M \gt 0$ there exists an N in the natural numbers such that for all ...
4
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3answers
78 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
3
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2answers
83 views

Probability of get a kidney donated

It is really a probability problem. I use the story of kidney donation because it is easier to describe. Consider the following scenario: Time is discrete. At each period, the measure of patients ...
1
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1answer
37 views

Convergence of a sequence with both sin and cos

I'm trying to figure out whether the following series converges absolutely or conditionally or whether it diverges. I am stuck on the following one that involved both sin and cosine: $$ ...
5
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3answers
108 views

An improper integral with parameter $\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$

I have to analyse the convergence of this integral: $$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in \mathbb{R}$. I have thought to write: ...
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5answers
84 views

Proof that $n(1-e^\frac{-\alpha}{n})$ converges to $\alpha$

I'm supposed to prove that $$\lim_{n\to\infty}n(1-e^{\frac{-\alpha}{n}})=\alpha.$$ It looks like it's supposed to be super easy but I'm stumped. Any suggestions?
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22 views

Conceptual question regarding convergence and continuity

Let $X,Y$ be metric spaces. Let $(f_n)$ be a sequence of functions from $X$ to $Y$ equicontinuous that converges pointwise to a function $f:X \to Y$. Then, $\{f,f_1,f_2,...\}$ is equicontinuous. ...
3
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1answer
20 views

Show convergence of sequence with infinite series as inequality

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics so Lipschitz hasn't passed the course yet. I should be able to prove this ...
2
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0answers
48 views

If $0\le x_{n+m}\le x_n\cdot x_m$, show that $\lim (x_n)^{1/n}$ exists [duplicate]

Suppose there is a sequence $\{x_n\}_{n\in\Bbb N}$ s.t. $$0\le x_{n+m} \le x_n\cdot x_m\quad\forall m,n\in\Bbb N$$ show that $$\lim_{n\to\infty}\left(x_n\right)^{1/n}=\xi\in\Bbb R$$ ...
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72 views

$\sum_{n=1}^{\infty} a_n$ converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \ge 1 $ ; then $a_n=0 , \forall n \in \mathbb N $? [duplicate]

Suppose that the series $\sum_{n=1}^{\infty} a_n$ of real terms converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \in \mathbb N $ , then how to prove that $a_n=0 , \forall n \in ...
7
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2answers
109 views

Taylor series not converging, other example than $\exp(-1/x^2)$?

The usual example for non-converging Taylor series is $g(x) = \exp(-1/x^2) \; \forall x \neq 0, g(0) = 0$: the Taylor series around $x=0$ is zero, but $g$ isn't zero for any $x \neq 0$. What's not so ...
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41 views

Find $\lim\limits_{n\to\infty}y_n$ if $y_1=\frac{x}{2},y_n=\frac{x}{2}+\frac{y^2_{n-1}}{2},0\le x \le 1,n=2,3,…$

Is it a good approach to use induction? If $0\le x \le 1$ then $0\le y_1 \le \frac{5}{8}$. Suppose that $$0\le y_n \le \frac{5}{8}$$ and prove $$0\le y_{n+1} \le \frac{5}{8}$$ If ...
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1answer
20 views

Counterexample for statements regarding to Raabe and convergence

Is there seq $\{a_n\}$ which absolutely converges and satisfies following condition: For any given $L>1$, there exists $n_{L}\in \mathbb{N}$ such that $1-1/n \geq |a_{n+1}/a_{n}| \geq 1-L/n$ for ...
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2answers
39 views

Is pointwise convergence useful?

The properties such as boundedness, continuity, integrability require uniform convergence of sequence of functions. I wonder, is there any property that pointwise convergence manages to transfer to ...
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1answer
33 views

The convergence of the power series $\sum \limits^{\infty }_{n=1}a_{n}(x-2)^{n}$ for various $x$

I would ask for help on how to solve this problem more specifically to know how to test whether a given $x$ converges in a power series. I would appreciate your insights. Of the power series $\sum ...
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0answers
19 views

What are the “classical ways” to study a vectorial sequence defined by a recurrence relation

Suppose that $F$ is a smooth (at least continuously differentiable) function defined from $\mathbb{R}^n$ to $\mathbb{R}^n$. The target is to study the recurrence sequence $X_{n+1}=F(X_n)$ when $X_0$ ...
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2answers
29 views

Ratio test for infinite products?

Is the ratio test applicable for testing convergence of infinite products? In other words, consider the sequence $(a_i)_{i=1}^\infty$ of non-zero real numbers. Also, consider the product ...
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2answers
32 views

Prove that $\lim\limits_{n\to\infty}y_{n}=\frac{1}{x}$ if $ y_{n}=y_{n-1}(2-xy_{n-1}), y_{i}>0,(i\in \mathbb{N}),x>0$

I have been asked to prove that $\lim\limits_{n\to\infty}y_{n}=\frac{1}{x}$ if $ y_{n}=y_{n-1}(2-xy_{n-1}), y_{i}>0,(i\in \mathbb{N}),x>0.$ In particular, what I would like to know is if it is ...
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2answers
44 views

Proof convergence of series

I'm trying to proof the convergence of the series $\sum\limits_{n=1}^{\infty} \exp\left(- \dfrac{n^k}{\log(n)} \right)$ where $0 < k < \frac{1}{2}$ is a positive constant. I tried to use the ...
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2answers
15 views

Sequence of convex non increasing sets convergence

I have a question for you. I was wondering whether a non increasing sequence of convex set converges to a convex set. Here my question made more precise: Let $\{S_k\}_{k=1}^\infty$ be a sequence of ...
1
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1answer
27 views

Proving a sequence is Cauchy (and convergent) by an infinite geometric sequence (something also with Lipschitz)

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics so Lipschitz hasn't passed the course yet. I should be able to prove this ...
0
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1answer
58 views

Proof that taylor series converges to function using taylors inequality

I would like to proof that the function $f(x)=\frac{1}{\sqrt{1-x}}$ converges to its Maclaurin series $$Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n$$ for $0<x<1$ by using taylors ...
0
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1answer
54 views

Distributions corresponding to $\frac{1}{|x|}$

Stirchartz's book ("A guide to distribution theory and fourier transforms" ) has Chapter 1 exercises Here $\mathcal{D(\mathbb{R}^1)}$ is a set of test functions $\phi:\mathbb{R} \rightarrow ...
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3answers
44 views

Is this what Convergence/Divergence comes down to?

In trying to understand why $\sum\limits_{k=1}^{\infty} \frac{1}{2^k}$ converges but $\sum\limits_{k=1}^{\infty} \frac{1}{2k}$ doesn't, I noticed that in infinite series of the type ...