Convergence of sequences and different modes of convergence.

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Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
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0answers
5 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
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0answers
39 views

How to prove that the series converges?

Let us suppose that $\vert a_n\vert$ is a real sequence, and set $S_n=\sum_{k=1}^{n}a_k$ and $\sigma_n=\frac{1}{n+1}\sum_{k=1}^{n}S_k$. How can we show that if the series $\sum_{n=1}^{\infty}\vert ...
2
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1answer
36 views

Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
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1answer
25 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
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1answer
36 views

Can someone help me prove that this sequence converges? [on hold]

I'm having difficulty trying to prove this, mostly because I don't understand the process. There was a proof similar to this in my textbook where they proved if Sn converges to s then lim 1/Sn=1/s and ...
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1answer
17 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
2
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1answer
17 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
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1answer
27 views

What is the solution to poissons equation with a point source?

I am currently trying to solve the following PDE using various numerical methods - direct and iterative. $$ \nabla ^2 u = \rho $$ where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The ...
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27 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
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25 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
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2answers
32 views

Convergence of sets implies containment [on hold]

Let l$_1$ be a set of {x$_n$} such that $\Sigma$$|x_n|$ is convergent. Let l$_2$ be a set of {x$_n$} such that $\Sigma$$|x_n|$$^2$ is convergent. So: l$_1$= {{x$_n$}$\in$R: $\Sigma$$|x_n|$ l$_2$= ...
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1answer
33 views

What is the difference between the limit of a sequence and a limit point of a set?

I always thought they were the same thing. The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point ...
2
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1answer
45 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
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3answers
31 views

Can someone help me understand the proof that every cauchy sequence is bounded?

This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I ...
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2answers
74 views

Don't understand proof that $x_n \rightarrow A$ $\iff$ every subsequence of $\{x_n\}$ converges to $A$

So we are given that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, we have that $|x_n - A| < \epsilon$ and we want to show that for all $\epsilon' > 0$, there ...
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1answer
40 views

Does the sequence $( n^{1/n} -1)$ belong to any $\ell^p$ space? [on hold]

The sequence $( n^{1/n} -1)$ converges to zero but does this sequence belong to the $\ell^p$ space for $p\in\mathbb R$? I don't know the answer, or how one would prove it. Same problem with the ...
2
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1answer
84 views

A step in proving that a real Cauchy sequence is convergent.

I'm trying to prove that a real Cauchy sequence is convergent, but I need some help for a step. We have the following statements: $\{ s_i\}$ is a real Cauchy sequence, i.e. ...
0
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1answer
17 views

uniform convergence of diff'ble functions with derivatives converging in $L^1$

Suppose we have a sequence of differentiable functions $f_n$ from a closed interval $I\subseteq\mathbb{R}$ to $\mathbb{R}$ with the following properties: $f_n$ converges uniformly to some function ...
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5answers
122 views

Prove that $c_n = \frac1n \left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \right)$ converges

I want to show that $c_n$ converges to a value $L$ where: $$c_n = \frac{\large \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}}{n}$$ First, it's obvious that $c_n > 0$. I ...
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0answers
18 views

Correct definition for convergence of a subsequence?

I only have the definition for convergence of a sequence, but can't find a definition for convergence of a subsequence. I have two guesses: For all $\epsilon > 0$, there exists an $N \in ...
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2answers
67 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
3
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1answer
48 views

Alternating infinite sum

I have the following infinite sum: $$ \sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}} $$ Because there is a $(-1)^n$ I deduce that it is a alternating series. Therefore I use the alternating series ...
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0answers
51 views

Don't understand proof that if $\{x_n\}$ is Cauchy and if some $x_{n_k} \rightarrow x$, then $x_n \rightarrow x$

So by definition of Cauchy, for all $\epsilon > 0$ and $i, j \in \mathbb{N}$, there exists an $M$ such that for all $i, j \geq M$, then $|x_i - x_j| < \epsilon'/2$ if we let $\epsilon = ...
0
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1answer
37 views

Don't understand proof that convergence implies Cauchy

So we are given that $x_n \rightarrow x$, so we can let $\epsilon = \epsilon'/2$ and there definitely exists an $N$ such that for all $n \geq N$, $|x_n - x| < \epsilon'/2$. Also by the triangle ...
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1answer
48 views

Sum of logarithmic series

Let $x_1>0$ we define sequence $(x_n)$ with formula $x_{n+1}=-\ln(x_1+x_2+\cdots+x_n)$ Find sum of the series $\sum_{n=1}^\infty x_n$. How to deal which such summation with logarithms?
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2answers
11 views

Convergence depending on the parameter

Let $c\ge 0$ be a real number. Then we define $$a_1=1, \quad a_{n+1}=\frac{cn+1}{n+3} a_n$$ Investigate convergence of $\displaystyle \sum_{n=1}^{\infty} a_n$ depending on the parameter $c$. Here I ...
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2answers
27 views

Investigate convergence of a series [on hold]

$$\sum_{n=2}^{\infty} \left(2\cdot(-1)^{\frac{n^2(n-1)}{2}}-1\right)\cdot\frac{1}{2n-7\sqrt{n}}$$ I need to investigate convergence of this series, and I have no idea which criterion will be the most ...
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1answer
34 views

Convergence of a series and sum [on hold]

I have the followin series: $\sum_{n=2}^\infty\left(\sum_{k=2}^n(-2)^{-k}3^{-n+k}\frac{1}{k!}\right)$ I need to investigate convergence of the series and calculate its sum. How to do this (both)? In ...
0
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1answer
43 views

Prove that if {$a_n$ } is a sequence that converges to A and $a_n$ ≥ 0 for all n then A ≥ 0.

Prove that if {$a_n$} is a sequence that converges to A and $a_n$ $\geq$ 0 for all n, then A $\geq$ 0. I have assumed on that contrary that A < 0. Pick $\epsilon$ = |A| > 0. Now, |$a_n$ - A| = ...
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1answer
47 views

Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
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2answers
49 views

Which test to choose for these series and why?

Which convergence test we need to choose for these two equations? $$\sum_{k=1}^{\infty}\frac{k}{10+k^{2}}\tag{1}$$ $$\sum_{k=1}^{\infty}\frac{1\cdot3\cdot5\cdots(2k+1)}{4^{k}\, k!}\tag{2}$$ For ...
4
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1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
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0answers
35 views

Uniform and absolute convergence of $\frac{1}{n^2+z^2}$

Let $z \in \mathbb{C}.$ I am asked to prove that $\sum\limits_{n=0}^{\infty} \frac{1}{n^2+z^2}$ converges on the set $\mathbb{C} \backslash \{ni : n\in \mathbb{Z}\} $. And also to prove that the ...
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0answers
30 views

sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
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1answer
31 views

Is this true?: converging sequence question

Let X be a metric space, and let A ⊂ X. Suppose that {pn} is a sequence in A which converges to some point p ∈ X. True or false: i) p ∈ A′ (limit points of A) (ii) p ∈ closure(A) These are both true, ...
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1answer
81 views

Prove $\lim_{n\to\infty}[(n^2+n)^{1/2}-n]= 1/2$

So I know that this for all epsilon>0, there exists an N such that for all n>N dp$ (p_n,\frac{1}{2})<epsilon $. But in this particular example I'm having difficulty finding the $N$ for which this ...
2
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1answer
31 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
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1answer
52 views

Problem 3.14(e) in Baby Rudin

If $\{s_n\}$ be a sequence of complex numbers, define its arithmetic mean $\sigma_n$ by $$\sigma_n \colon= \frac{s_0 + s_1 \cdots + s_n}{n+1} \, \, (n = 0, 1, 2, \ldots). $$ Put $a_n = s_n - s_{n-1}$ ...
4
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1answer
68 views

Convergence in measure implies convergence almost everywhere (on a countable set!)

Here is an interesting problem from "Real Analysis for Graduate Students" by Richard Bass (which is an amazing book, by the way). Suppose $(X, \mathcal{A}, \mu)$ is a measure space, and $X$ is a ...
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1answer
17 views

Convergence of sums of indicator random variables

I'm working through some practice problems for my final exam and I would like to get some ideas on tackling this problem: Let $(\Omega,\mathcal{F})=(\mathbb{R}_+,\mathcal{B}(\mathbb{R}_+))$ and ...
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2answers
40 views

Convergence of a certain series of Primes

This is a problem from Alan Baker's Comprehensive Course in Number Theory. We have to show that $\displaystyle \sum\limits_{p} \frac{1}{p (\log\log p)^{\delta}}$ converges for all $\delta >1$.Here ...
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1answer
13 views

Independent random variables convergence almost surely

I'm preparing for a final exam and I'm working through a practice exam. I'm somewhat stuck on a problem. The random variables $\{X_i\}$ are all independent and all satisfy $E[X_i^4]\leq 1.0$, but ...
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2answers
41 views

For what sequences $a_n$ does the sequence $(1+\alpha a_n)^n$ converge?

We know $ (1+\alpha/n)^n \rightarrow e^{\alpha} $ when $n\rightarrow +\infty$. Suppose we are given a modified version of the problem: $$ \quad (1+\alpha\cdot a_n)^n \tag{1} $$ The question ...
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1answer
80 views

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$ implies $\lim_{n \to \infty} \int_B f_n \, d\mu = \int_B f \, d\mu$ for $B \subseteq X$

I'm having trouble with the following problem. Let $(X, \mathcal{M},\mu)$ be a measure space, where $X = [a,b] \subset \mathbb{R}$ is a closed and bounded interval and $\mu$ is the Lebesgue measure. ...
3
votes
1answer
52 views

Does the sequence $f\chi_{E_n^c}$ converge pointwise to $f$ if the measure of $E_n$ tends to zero?

If $(X,\mathcal{M},\mu)$ is a measure space, $f:X\rightarrow \mathbb{R}$ are such that $f\geq 0$ and $E_n\subseteq X$ measureable are such that $\mu(E_n)\rightarrow 0$, how would I be able to show ...
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3answers
39 views

convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
4
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2answers
89 views

Study the convergence of $\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$

Can you help me to study the convergence of the following series: $$\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$$ Thanks.
0
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1answer
12 views

Infinite sequence of real numbers converging to x and y

So the question is: Suppose $x_i$ and $y_i$ are infinite sequences of real numbers converging to x and y. Show that $(x_i + y_i)$ converges to $x+y$. Show that $x_iy_i$ converges to $xy$. Here's ...
0
votes
1answer
29 views

Practice with closed sequences

Okay I have a final coming up, but I realized that I am still not adept at writing proofs. *Prove that if $\lim_{n\to \infty}p_n = p$ in a given metric space then the set of points {$p, p_1, p_2, ...