Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
2answers
25 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
0
votes
1answer
13 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence- an = ln(n)/(n^(1/n)) is divergent. Can anyone help me out with this? Thanks!
0
votes
0answers
29 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
0
votes
0answers
12 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
2answers
25 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
1
vote
1answer
24 views

Prove that the sequence $(b_n)$ converges

Prove that if $(a_n)$ converges and $|a_n - nb_n| < 2$ for all $n \in \mathbb N^+$ then $(b_n)$ converges. Is the following proof valid? Proof Since $(a_n)$ converges, $(a_n)$ must be bounded, ...
0
votes
0answers
12 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: ...
0
votes
1answer
52 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
1
vote
0answers
24 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
24 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 ...
1
vote
1answer
21 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, ...
0
votes
0answers
20 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 ...
-1
votes
2answers
37 views

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

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 ...
2
votes
0answers
26 views

Convergence in probability, expected value

I have problems with the following two sequences of random variables: We assume that $X_1, X_2, ... $ are iid. Let $m=EX_i$ The first one is: $$ \alpha_n := \frac{1}{n} \sum_{i=1}^n (X_i - m)^2$$ I ...
0
votes
1answer
32 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 ...
1
vote
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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
33 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 ...
2
votes
1answer
57 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 ...
1
vote
0answers
16 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 ...
-1
votes
4answers
57 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} ...
0
votes
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.
0
votes
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 ...
0
votes
1answer
11 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 ...
1
vote
2answers
53 views

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

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
0
votes
1answer
23 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
4
votes
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 ...
2
votes
1answer
48 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 ...
2
votes
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 ...
1
vote
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? $$d_n=\left(1+\frac{2}{n}\right)^n$$
0
votes
2answers
30 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 ...
2
votes
1answer
32 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 ...
0
votes
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 ...
0
votes
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: ...
0
votes
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. ...
2
votes
0answers
26 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 ...
3
votes
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. ...
1
vote
3answers
239 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 ...
1
vote
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 ...
1
vote
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 $$\int_{a}^{b}f(x)dx$$ ...
-3
votes
2answers
29 views

How to prove if a series converge

I need to prove if this series converge and if yes, prove the partial sum. I have no idea where to start Can somebody tell me the steps i need to follow?
0
votes
5answers
67 views

Determine whether the series $\sum_n \frac{1}{n^n}$ converges or not? [closed]

I believe I need to use the ratio test to prove it is convergent.
0
votes
2answers
24 views

Convergence of a sequence of matrices

Let $A$ be a $n×m$ matrix with real entries, and let $B = AA^ t $and let $\alpha$ be the supremum of $x ^t Bx$ where supremum is taken over all vectors $x ∈ \mathbb R ^n$ with norm less than or equal ...
0
votes
1answer
21 views

The convergence set of a sequence of functions can be expressed in terms of upper and lower envelopes

let $f_n:\mathbb R\to[0,\infty)$ be a sequence of functions. Its lower envelope sequences are defined as $\underline{f_n}(x)=\inf\{f_k(x):k\geq n\}$. And its upper envelope is defined similarly except ...
4
votes
2answers
55 views

If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if ...
1
vote
3answers
61 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 $x_0=1$ ...
2
votes
1answer
43 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 ...