Convergence of sequences and different modes of convergence.

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16 views

show that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? [on hold]

Prove that $f_n\to 0$ (lebesgue measure), is not convergent in any point of $x \in [0,1]$? Hint: let $f_n$ be the characteristic function of n-th interval in: ...
-1
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0answers
56 views

Convergent series and real numbers [closed]

Prove that every decimal representing a positive real number can be expressed as a convergent series. Any ideas?
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0answers
49 views

Prove the uniform convergence

Let $a_n$ be a monotonic sequence convergent to a. Let f : R $\to$ R be a continous and monotonic function. Then we define a series of functions as follows : $$f_n(x) := f(x+a_n)$$ Prove that the ...
0
votes
1answer
26 views

Euler method uniform convergence

I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's ...
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1answer
24 views

Almost sure convergence, arithmetic mean, variance

I am stuck proving that this sequence $$\sigma^2_n:= \frac{1}{n-1} \sum_{i=1}^n (X_i - \frac{X_1+...+X_n}{n})^2$$ is converegent almost surely to $D^2X_i = \sigma^2$. We assume that $X_1, X_2, X_3, ...
2
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1answer
82 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
0
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0answers
21 views

Interchanging infinite double sum and expectation

Let $\xi_i$ be a sequence of independent and identically distributed standard normal random variables and consider sequences $\{b_i\}$ and $\{c_j\}$ such that $\sum_i b_i<\infty$ and $\sum_j ...
8
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2answers
3k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
-1
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1answer
40 views

Give an example of a divergent and a convergent series such that the following holds: [closed]

I'm having trouble with this: I need to find an example of a divergent series $\sum_{n=1}^\infty a_n$ of positive numbers $a_n$ such that $lim_{n \rightarrow \infty }$ $a_{n + 1}/a_n$ = $lim_{n ...
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1answer
35 views

Puzzled at this alternating series problem.

I have rechecked this problem so many times, and even my tutor got stuck on this. Can someone tell me what I did wrong? My homework says I got at least one question wrong. And my tutor was confused ...
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1answer
27 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
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1answer
599 views

Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
2
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1answer
36 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
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1answer
34 views

Convergence of infinite series of function with factorial and power

Determine whether the series is convergent or divergent: $$\sum_{n=0}^\infty \frac{(3n)!+4^{n+1}}{(3n+2)!}$$ I guess we have to use comparison test for this question, but I am not sure what to use ...
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1answer
13 views

Asymptotic convergence of the total length of a graph

I encoded the following algorithm: suppose we're in (0,1)x(0,1) and I randomly create a "village" one at a time. At each step, I link a newly randomly created village to the closest village already ...
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3answers
77 views

How to prove that $\log\left(\frac{x+1}{x}\right) - \frac{1}{x+1}$ is always positive and tends to zero?

What I get so far is the inequality: $$1 + \frac{1}{x} > e^{\frac{1}{x+1}}$$ which if we expand: $$1+\frac{1}{x} > 1+\frac{1}{x+1}+\frac{1}{2! (x+1)^2}+\cdots$$ and I cannot prove this ...
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2answers
71 views

Proof of limit $\lim_{x\to0}\left(\frac{1}{x}\right)^x = 1$

How do you prove that the limit $$\lim_{x\to0}\left(\frac{1}{x}\right)^x = 1$$ I have tried doing this: Let$$y = \lim_{x\to0}\left(\frac{1}{x}\right)^x$$ $$\ln(y) =\lim_{x\to0} (x*(-\ln(x)))$$ ...
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4answers
59 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
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2answers
178 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. $$\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$$ Intuitively, I believe ...
2
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1answer
89 views

Almost sure convergence of the series of independent random variables

Let $\{X_n:n\ge1\}$ be i.i.d. random variables with $\operatorname EX_1=0$ and $\operatorname E|X_1|^p<\infty$, where $1<p<2$. Let $\{b_n:n\ge1\}$ be a real sequence. Does the series $$ ...
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0answers
18 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
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2answers
97 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
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1answer
21 views

Root Test for Convergence or Divergence (ln problem)

I'm stuck at this problem. How would I approach this (I have to use the root test for this one)? The ln and e and my power is throwing me off. This is how far I have gotten.
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1answer
15 views

Interval of convergence for a series

I am currently trying to determine the interval of convergence, but I keep getting 0 for all my questions. I have attached one of the questions that I am unable to solve completely and I would really ...
9
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3answers
262 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
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2answers
53 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [closed]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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2answers
166 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
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1answer
24 views

Almost uniform convergence implies a.e. pointwise convergence proof

I've just read a proof of the statement "On a finite measurable space, $(f_n)_{n \geq 1}$ and $f$ measurable and finite a.e. functions, if $(f_n)_{n \geq 1}$ converges almost uniformly to $f$, then it ...
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2answers
21 views

Use ratio test to test for convergence or divergence

I have online hw and it tells me if my answer is correct or not. It said that my answer for this problem is incorrect: Can someone tell me what I did wrong? Also I might be asking alot of these ...
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1answer
41 views

Showing convergence and divergence

Say I have: $(x_n)$ a sequence of real numbers such that $\sum x_n$ which converges conditionally and implies $\sum x_{2n}$ diverges. I want to show that $x_{2n}$ does not in general converge. So I ...
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2answers
43 views

Understanding complex numbers

I need to show that $$\left | \sum_{k=1}^n e^{ik}\right | $$ is bounded Now I am given that $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ But have little idea of how to proceed further and ...
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5answers
2k views

Is there a way of working with the Zariski topology in terms of convergence/limits?

As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...
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2answers
41 views

How to prove that a sequence converges

I am having some trouble understanding how I can show that a given series converges. I found a general explanation here that states: To prove that a sequence converges, it is sometimes easier to ...
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1answer
31 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
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1answer
49 views

Question about convergence in $L^2$

Assume we have a sequence of functions $\{f_n\}_{n\geq 0}\subset L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$, i.e. $$\lim_{n\to \infty} \int_0^1 |f_n(x)-f(x)|^2dx =0.$$ Is it then true ...
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1answer
61 views

Exponential of a matrix always converges

I am trying to show that the exponential of a matrix converges for any given square matrix of size $n\times n$: $M\mapsto e^M$ e.g. $\displaystyle e^M = \sum_{n=0}^\infty \frac{M^n}{n!}$ Can I argue ...
2
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1answer
47 views

Does $\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$ converge?

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$$ Because this is an alternating series, I decided to use the alternating series test. This ...
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4answers
41 views

$d_n=(1+(2/n))^n$ converge or diverge and find the limit?

I know the answer is $e^2$ and I'd like to use L'Hopital's rule because this is an indeterminate form. Can someone explain how to get there? $$d_n=\left(1+\frac{2}{n}\right)^n$$
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1answer
34 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...
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2answers
33 views

Convergence of infinite series from 2 to infinity 1/(x((lnx)^2))

On a recent exam I was asked to test the following series for convergence From $2$ to $\infty$ $\frac{1}{x(lnx)^{2}}$ I blanked on the integral but set up a comparison test, saying that ...
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1answer
313 views

Convergence in normed spaces

I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$. Does $Q_nf_n$ ...
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0answers
22 views

Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
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0answers
14 views

How would I prove the following convergence? [duplicate]

I found the following code in an algorithms book and can't really seem to prove why it returns a square root, I understand it intuitively, but can't prove it. ...
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1answer
45 views

Riemann-Stieltjes Integrability and Convergent Series

Let $\alpha_{n=1}^{\infty}$ be a sequence of monotonically increasing functions on $[a.b]$ such that the series $\sum_{n=1}^{\infty}\alpha_{n}(a)$ and $\sum_{n=1}^{\infty}\alpha_{n}(b)$ converge. ...
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0answers
27 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
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3answers
62 views

Convergence of sequence $x_{n+1}=\sqrt{2x_n+3}$

Let $f:\Bbb R \rightarrow \Bbb R$ be the function $f(x)=\sqrt{2x+3}$. (a) Show that for all $x\in[1,3], f(x)\geq x$. You may use the intermediate value theorem if you like. Let $x_0=1$ ...
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3answers
241 views

Absolute convergence to zero imply convergence to zero?

Given that $\frac{1}{N}\sum_{i=1}^N {|a_i|}$ converges to zero as $N\rightarrow \infty$, does it imply that $\frac{1}{N}\sum_{i=1}^N {a_i}\rightarrow 0$? I know absolute convergence imply ...
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2answers
15 views

Conditionally convergent sequences and implications

If I have $\sum b_n$ is conditionally convergent, how can I show that $\sum b_{4n}$ doesn't in general converge? Assume $(b_n)$ is an arbitrary sequence of the Reals All I need is a counter example ...
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1answer
27 views

proof related to convergence of a integral

i have the following condition $$0\le f(x)\le g(x)$$ and $$\int_{a}^{b}g(x)dx$$ is convergent for any $a$ and $b$ (which means $a$ or $b$ can tend to infinity) then prove that $$\int_{a}^{b}f(x)dx$$ ...
2
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1answer
44 views

Convergence to infinity of a sum of independent random variables

I am doing an exercise which says: Suppose $(X_n)$ is a sequence of independent random variables (not necessarily identically distributed) with finite variances. Write $S_n:= \sum_{i=1}^n X_j$ for ...