Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

1
vote
1answer
71 views

Determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$

How to determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$ ? I tried to get somewhere with Integral criteria and with comparing to other series but ...
3
votes
2answers
45 views

When are both $\sum_{n=0}^\infty \log(a_n)$ and $\sum_{n=0}^\infty a_n$ convergent?

I'm new to this site. Can someone give me some examples of when both: $$\sum_{n=0}^\infty \log(a_n)\qquad \text{ and }\qquad \sum_{n=0}^\infty a_n$$ are convergent?
3
votes
1answer
44 views

Can anyone prove D'Alembert Criterion (Dalambert) criterion for converging positive sequences?

This will most likely be on the exam, but it is not given in the text book. In my notebook I have this proof which I will type out, but it makes no sense. Here it goes: $$\text{D'Alembert ...
2
votes
0answers
178 views

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
2
votes
1answer
28 views

$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
1
vote
1answer
34 views

Convergence rate of a function

I'm having a difficult time working out the details of the following problem. I'm hoping someone may be able to point me to a reference or suggest an approach. I have three matrices $(A, B, C)$ and ...
1
vote
1answer
27 views

Convergence of subsequence of partial sums implies full convergence?

Define $S_n = \sum_{j=1}^n a_j$ a sum of real numers. Suppose there is a subsequence such that $S_{n_j} \to S$ for some $S$, i.e., the infinite sum converges along this subsequence. Does this imply ...
0
votes
2answers
62 views

dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n ...
0
votes
2answers
47 views

Almost sure convergence of $\max(X_1, X_2,\ldots,X_n)$.

$X_1, X_2,\ldots, X_n$ are independent with uniform distribution on $[0,a]$. Prove that $\max(X_1, X_2,\ldots,X_n) \rightarrow a$ almost surely. Ar first I look for the probability distribution i.e. ...
1
vote
0answers
36 views

Limit of integrals of simple functions over a finite measure

We are given a sequence of simple functions $f_n:\mathbb{R}^2\rightarrow \mathbb{R}$ which converge pointwise to a continuous limiting function $f$. We also have a bunch of positive, finite measures ...
1
vote
1answer
28 views

How do I find the interval of convergence?

Suppose I have: $$\sum \cfrac{(-1)^n}{\sqrt{n}}x^n$$ If I use the ratio test, I get $$\cfrac{1}{\sqrt{1+\frac{1}{n}}}|x|$$ Why can it be said the radius of convergence here is $1$? Disregard the ...
0
votes
0answers
16 views

Limits and Convergence of sequences in the form of $(k, k^2, 1/k)$

I'm dealing with proving the convergence and limits of sequences that are defined by multiple points, such as $$ \left(k, k^2, \frac{1}{k}\right) $$ and I'm not sure how to go about doing it. I'm ...
6
votes
0answers
177 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
8
votes
4answers
571 views

Integral of odd function doesn't converge?

When I look up $$\int_{-1}^1 \dfrac{1}{x} dx$$ on Wolfram Alpha, it says it doesn't converge. While this is a sum of two diverging integrals, the two areas are clearly symmetric, and I'd assume the ...
0
votes
1answer
29 views

Convergence of Series for tangent (only convergence or divergence)

$$\sum_{n=17}^{\infty}\left(\tan\left(\frac{1}{n}\right)\right)^2 \ \ $$ My first guess is to write the series as integral. And use the substitution for u=1/n. That changes my upper and lower ...
1
vote
1answer
44 views

Prove completeness of a metric space

Let $\mathcal{K} = \{A \subset \mathbb{R}^N| A \neq \emptyset, A \text{ closed and bounded with respect to the euclidean metric} \}$ Let us define $A_\epsilon = \bigcup_{x \in A}U_\epsilon(x)$, where ...
0
votes
2answers
57 views

Find radius of convergence for the given sequence: $\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$

I've been trying to realize how to find the radius of convergence for this sequence: $$\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$$ I know that it converges for any given $x$, but can someone explain me ...
5
votes
2answers
62 views

Show that the sequence $\langle b_n\rangle$ Converges to $1$

The following question was asked in my Masters entrance examination but unfortunately I was unable to answer this. Please tell me how to approach this problem correctly. Suppose $\langle ...
0
votes
0answers
17 views

correlated random vectors converge to an independent random vector

Suppose $(X_k,Y_k)_{k\geq 1}$ is a sequence of random vectors. $X_k,Y_k$ have support $[0,1]$ and a joint density function $ f_k(x,y)$ being positive for all $x,y\in [0,1]$ and continuous everywhere. ...
2
votes
3answers
69 views

Integral test for convergence of $\frac{1}{\ln x}$ [closed]

I want to know if $$\int_0^1 \frac{1}{\ln x}\, dx$$ converges or not.
5
votes
3answers
65 views

Determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$

I tried to use D'Alambert theorem to determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$ . $\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} ...
0
votes
1answer
53 views

Determine convergence of the series $\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$

How to determine convergence of the series: $$\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$$ I spent most of the time using the Integral criteria (since the function $f(x)=\frac{1}{\ln(x)^{\ln(x)}}$ ...
1
vote
1answer
26 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n ...
2
votes
3answers
31 views

Conditionally convergent - limit of series'

Let $\sum_{n=1}^\infty a_n$ be conditionally convergent. Let $k_n:= \max(a_n,0),l_n:=-\min(a_n,0)$ for $n\in \mathbb{N}$ and show that $\sum_{n=1}^\infty k_n =\infty $ and ...
2
votes
0answers
34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
0
votes
0answers
15 views

Rearranging series' to converge to a certain point.

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
0
votes
0answers
14 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
0
votes
3answers
33 views

Trouble with the proof of convergence of a series.

$\sum_{n=1}^\infty \frac{n}{(-2)^n}$ I tried using D'Alembert's Ratio on it and this is how far I got: $\frac{(n+1)}{(-2)^{n+1}}\frac{(-2)^n}{n}=\frac{n+1}{(-2)\cdot ...
1
vote
1answer
28 views

Convergence in probability implies boundedness in $L^1$?

Suppose we have a sequence of positive random variables wgich converges to 0 in probability, i.e. $X_n=o_P(1)$. I want to prove that $E[X_n]$ is bounded. My idea: In particular $X_n$ is bounded in ...
3
votes
1answer
67 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
0
votes
4answers
74 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
0
votes
1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
0
votes
1answer
29 views

Snowflake-sequences - Area - Circumference

Consider the following inductively defined snowflakes-sequences: $S_1$ is an equilateral triangle with edge length $l_0$, and $S_{n+1}$ emerges from $S_n$ by dividing each edge by 3 and the middle ...
1
vote
2answers
39 views

Series' - Convergence - Limit of sequences

Examine the following series' for convergence: a)$\sum_{n=1}^\infty \frac{n^3\cdot 3^n}{n!}$, b)$\sum_{n=1}^\infty\frac{n}{(-2n)^n}$, c)$\sum_{n=1}^\infty \frac{n!}{n^n}$, ...
1
vote
3answers
42 views

Convergence of the series $\sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right)$ [closed]

Does the series $$ \sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right) $$ converge?
2
votes
4answers
95 views

How would you prove this converges? $\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$

How would you check if this converges or not? $$\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$$ It looks like a telescopic sequence so I thought I'd first write the beginning values: ...
5
votes
4answers
152 views

How to prove that $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$

I have this sequence: $$\sum_{1}^{\infty} \frac{1}{n^3}$$ and I need to prove that: $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$ So basically I know that this sequence converges using the integral ...
0
votes
1answer
33 views

Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
-1
votes
1answer
101 views

prove the convergence sequence [closed]

How to prove that the sequence is convergent? ${a_{n+2} = {\sqrt{a_{n+1}} + \sqrt{a_{n}}}}$ , where $a_{1}=1$ and $a_{2}$ is a positive number.
3
votes
1answer
25 views

Example of non-commutative infinite product of complex numbers.

I have read a proof of the following theorem in Rudin's Real and Complex Analysis: Suppose $\{u_n\}$ is a sequence of bounded complex functions on a set S, such that $\sum |u_n(s)|$ I converges ...
1
vote
2answers
58 views

How to solve this sequence?

I have this sequence: $\sum_{n=1}^{\infty} \frac{n^2+n-1}{\sqrt{n^\alpha+n+3}}$ For which values of $\alpha$ does this converge? I first tried to separate into cases where $\alpha \gt 0$ etc and ...
3
votes
1answer
58 views

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $u\in C^0(\Omega)$ and $(u_n)_{n\in\mathbb{N}}\subseteq C_0^0(\Omega)$ with $$\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0\tag{1}$$ Can ...
0
votes
2answers
50 views

product converging to one

I have a question concerning an infinite product. Suppose $x_n$ is a sequence of positive real numbers. My intuition says that $$\lim_n(1-\exp(-x_n))^n=1$$ for any sequence $x_n=n^\alpha$ with $\alpha ...
2
votes
1answer
40 views

Show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s.

Given a sequence $(X_n)_{n\geq 1}$, show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s. Here is my attempt: $|X_n|\leq Y$ a.s. means that $P(|X_n|>Y)=0$, $\forall n\geq 1$ $P(\sup_n ...
2
votes
1answer
36 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
0
votes
2answers
33 views

How can I use the sequential criterion to prove/disprove the existence of a limit?

In my book (Abbott), it is written that the sequential criterion for the limit of a sequence is as much of a tool to prove as it is to disprove the existence of a limit. My question is: how? For ...
3
votes
1answer
71 views

Examples and counter-examples in Real analysis - check my answers please

I've been given some practice examples, without solutions in preparation for an upcoming exam, and was hoping I could get them double checked here. For each of the following, either give an example ...
0
votes
0answers
30 views

How to interpret this statement about two sequences?

Say, $(a_n)_n, (b_n)_n$ are two sequences of non-negative real numbers and both converge to zero. Moreover, we know that $$\forall \varepsilon,M > 0 \ \ \ \exists n_0 \in \mathbb N \ \ \ \forall n ...
0
votes
0answers
21 views

Verifying Integrals with convergence theorems

I would like to check with you guys, whether my solutions are correct so far. Problem: Verify following statements $\lim\limits_{n\rightarrow\infty} \int_0^\infty e^{-nx} \sin(e^x) dx = 0$ ...
2
votes
1answer
91 views

Verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0.$

How to verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0$? My idea is to use the dominant convergence theorem with $f_n(x):= e^{-nx} \sin(e^x)$ and ...