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Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

3
votes
2answers
33 views

Why does this sum converge: $\sum_{k=2}^{n-1} \frac{1}{k} \frac{n \mod k}{n}$

I was playing around with numbers and wanted to create a function that somehow indicates if a number could be a prime. So I came up with this, with the intention that it should make small jumps if $x$ ...
1
vote
2answers
31 views

Show using the definition of convergence that the sequence 1/n does not converge to any number $L > 0$.

Let $\epsilon > 0$ $|1/n - L| < \epsilon$ for some $n\geq N$. (definition of convergence) This implies $-\epsilon\cdot n < 1 - nL < \epsilon\cdot n$ Therefore $1 - nL > -\epsilon\...
0
votes
0answers
38 views

What are some series whose convergence is not known

I found series involving division by powers of sin whose convergence is not known. But what about the following: 1) Are there series whose convergence does not depend on external unknown problems (...
2
votes
0answers
54 views

Convergence of the Power Series for $xu''+\sin(x)u=0$

Consider the initial value problem $$xu''+\sin(x)u=0 \ \ \ \ u(0)=0, u'(0)=2$$ What can be said about the radius of convergence of this series? I have determined that the first four nonzero terms ...
4
votes
3answers
79 views

Why $\sum_{n=1}^{\infty}\frac{n\left(\sin x\right)^{n}}{2+n^{2}}$ is not uniformly convergent on $[0,\;\frac{\pi}{2})$?

Why $\sum_{n=1}^{\infty}\frac{n\left(\sin x\right)^{n}}{2+n^{2}}$ is not uniformly convergent on $[0,\;\frac{\pi}{2})$? I was thinking that we need to show partial sums \begin{equation} \left|S_{2n}\...
2
votes
2answers
31 views

Is $f_{n}\left(x\right)=\sin\left(x+\frac{x^{2}}{n}\right)$ uniformly convergent on $\left[0,\:2\pi\right]$

Is $f_{n}\left(x\right)$ uniformly convergent on $\left[0,\:2\pi\right]$? \begin{equation} f_{n}\left(x\right)=\sin\left(x+\frac{x^{2}}{n}\right) \end{equation} We can see that $f_{n}\left(x\right)$ ...
0
votes
2answers
22 views

Why we ONLY use ratio test and not conditional convergence to determine the interval of convergence of an alternating series?

For example, consider $$S_n=\sum_{n=1}^{\infty} \frac{(-1)^n x^n} {\sqrt{n}}$$ While determining the interval of convergence, we use the ratio test to determine the interval in which the series ...
0
votes
1answer
14 views

About trigonometric test

We know that, if $b_k$ is monotone decreasing and $\lim b_k =0$, then $\sum b_k \sin kt $ is convergent for all $t\in R $. IF we change the condition from $b_1 \geq b_2 \geq ....\geq 0$ to $b_N+1 ...
-1
votes
0answers
13 views

Help with monotone convergence theorem (RV)

This little problem may be simple thing to an expert in the field but it drives me crazy. For a sequence of R.V., $0\le X_1\le X_2 \ldots \le X_n \ldots$, I have $\lim_{\rightarrow \infty} E[X_n] = ...
0
votes
1answer
54 views

What is the series expansion of the $n$-th derivative of this : $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\text{erf}(x)}dx$

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\erf(x)}dx$ with wolfram alpha we have for $n=1, n=2, ..n=4$ interesting expansion which seems present ...
0
votes
0answers
16 views

Convergence in Lp, exponential and maximum

I am having trouble with the following problem: Let $X_j$ be IID exponential dist. with mean $\theta>0$, and let $N_j$ be IID (and also independent of $X_j$) with probability mass function: $P(...
0
votes
1answer
47 views

convergence of Cauchy sequences defined recursively

$X=(x_n)$ defined as $X_1=1,X_2=2$ and $X_n=\dfrac12\left(X_{n-2}+X_{n-1}\right)$. To finds its limit, first we should prove that it is convergent, but my book has not given its solution. Instead, it ...
0
votes
4answers
55 views

Is $\sum_{n=1}^{\infty}\frac{\sin a_{n}}{\sqrt{n}+na_{n}}$ convergent?

Suppose $a_{n}>0$ and $\sum_{n=1}^{\infty}a_{n}$ is convergent. $\sum_{n=1}^{\infty}\frac{\sin a_{n}}{\sqrt{n}+na_{n}}$ convergent ? Since $\sin a_{n}$ is bounded by one, and $\sqrt{n}\...
19
votes
1answer
359 views

Does recursively replacing $\frac1n$ by $\frac1n(\frac12+\dots+\frac1{n+1})$ really converge to $\frac1e$?

I was thinking of a problem and have an answer through computer programming, but am unable to prove it mathematically. Start with the following: $$\frac{1}{2}\bigg(\frac{1}{2} + \frac{1}{3}\bigg)\...
0
votes
2answers
14 views

Difference between constant and random variable always equal to constans

I am interested in the difference between constant (let's call it $c$) and random variable which is always a constant: $P(X = c) = 1$. Is there any trap in thinking that it is the same? What about ...
2
votes
1answer
57 views

What method is most preferred to find nature of the series $\sum_{n=1}^{\infty} \frac{a^n}{a^n+x^n}$?

I tried to use D'Alembert's ratio test to test the convergence of the given series $\sum\limits_{n=1}^{\infty} \frac{a^n}{a^n+x^n}$ but I could only solve for $\frac{a}{x} <1$ and it is coming out ...
2
votes
1answer
30 views

a.s. convergence uniform distribution

I am having troubles with proving almost surely convergence for the following problem: Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be: $A_n=\sum_{k=1}^n \prod_{j=1}^k U_j$ for $n\in \...
-2
votes
4answers
57 views

Easiest way to show sequence $e^{\frac{1}{n}}$ converges to 1 [on hold]

I don't want to use the definition, is there an easier way? Like, in the case of limits of functions, it is pretty easy to say by the composition theorem. Is there any such theorem in case of ...
0
votes
0answers
18 views

Convergence results for conditional random random variables

Suppose that $X_n$ is a sequence of random variables such that $X_n \rightarrow X$ either almost surely or in $L^p$ or both. Now consider the sequence of conditional random variables $f(X_n) | X_n \...
2
votes
1answer
138 views

Convergence/Divergence of an Infinite Series with Natural Logarithms

I've spent a good week and half manipulating and trying different tests to find the convergence or divergence of this series: $$\sum_{n=0}^\infty \frac{1}{(\ln n)^{\ln n}}$$ I've tried all the ...
0
votes
1answer
31 views

Is $f\left(x\right)=nx^{n}\left(1-x\right)$ uniformly convergent on $x\in\left[0,\:1\right]$ [duplicate]

Is $f_{n}\left(x\right)=nx^{n}\left(1-x\right)$ uniformly convergent on $x\in\left[0,\:1\right]$ We can see $f_{n}\left(x\right)\rightarrow f\left(x\right)=0$ point-wise, but $f_{n}\left(x\right)\geq ...
2
votes
2answers
30 views

Convergence both in $L^p$ and $L^q$

I am trying to understand convergence in Lp spaces a little bit better. If we have $p, q \in [1, \infty]$ and a sequence of functions $(f_n)_{n\in\mathbb{N}} \subset L^p(\mathbb{R}^d) \;\cap\; L^q(\...
2
votes
1answer
47 views

Convergence of $\sum_{n=1}^{\infty}\left(\frac{\sin(2\alpha^n)}{\alpha^n}-2\right)$

Determine values of $\alpha \in \left[\frac{1}{2} ; \frac{3}{2}\right]=:I$ s.t. the following series converges: $$\sum_{n=1}^{\infty}\left(\frac{\sin(2\alpha^n)}{\alpha^n}-2\right)$$ Am I doing it ...
0
votes
1answer
33 views

Convergence of sequence of functions in metric spaces

Let $f_n(x)=\frac{nx}{nx+1}$ $a)$ Show that in $C_{[0,2]}$ $$\lim_{n \to \infty}{f_n(x)}=1$$ $b)$ Does $f_n$ converge in $C_{[0,1]}?$ Here is my attempt: $a)$ We need to show that $d(f_n,f)&...
0
votes
1answer
73 views

What's equal this:$\int_{0}^{\infty}\prod_{k=0}^{n}{\cot^{-1}(x^k+\frac{1}{x^k})}dx$?

Evaluation for some integer $k$ using wolfram alpha show that this integral $\int_{0}^{n}\prod_{k=0}^{n}{\cot^{-1}(x^k+\frac{1}{x^k})}dx$ converges fast and deacreasing to $0$ , Now my question ...
1
vote
1answer
33 views

What is the interval of convergence for power series

I have a power series and I need to find the interval of convergence. $$\sum_{n=0}^\infty= \frac {n}{(n^2-1) (1+x)^n} $$ I tried ratio test with a new variable $ t = \frac{1}{1+x} $and I get that ...
3
votes
0answers
45 views

Sum of two sequences converges and one of them is bounded implies they both converge

Let $(x_n), (y_n)$ be two real sequneces such that $z_n:=x_n-y_n\to l\in\mathbb{R}$. Prove that if $(x_n)$ is bounded, then $(x_n)$ and $(y_n)$ are convergent. I found this assignment in a sample ...
0
votes
2answers
31 views

Convergence of $\sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)}\,\,\,\,; k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)}$

The convergence of the following series as $a \in \mathbb{R}$ $$\sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)}\,\,\,\,; k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)}$$ As $n \to +\infty$ we ...
1
vote
3answers
66 views

Is $\sum_{n=1}^\infty \frac{n^2+1}{n!2^n}$ convergent? If yes, evaluate it.

Is $\sum_{n=1}^\infty \frac{n^2+1}{n!2^n}$ convergent? If yes, evaluate it. Using the ratio test, I could show that the series is convergent. What is an easy way to evaluate it? I was thinking about ...
2
votes
3answers
67 views

Convergence of $\sum \frac{\left|\sin\left(\left(1-\frac{1}{n}\right)^n-\frac{1}{e}\right)\right|^{\alpha}}{e-\left(1+\frac{1}{n}\right)^n}$

Study the convergence of the following series as $\alpha \in \mathbb{R}$ $$\sum_{n=1}^{\infty}\frac{\left|\sin\left(\left(1-\frac{1}{n}\right)^n-\frac{1}{e}\right)\right|^{\alpha}}{e-\left(1+\frac{1}{...
1
vote
1answer
47 views

Convergence in $L^p$ of $X_nY_n$ given $X_n$ and $Y_n$ converge

My definition for convergence in $L^p$ is the following; A sequence $x_n \to x$ in $L^p$ to a random variable $x$ if $|x|^p$ is integrable and $E[|x_n -x|^p] \to 0$ as $n \to \infty$ My question is, ...
1
vote
3answers
50 views

Is the following sequence convergent?

$$a_n=\prod_{i=1}^{n}\frac{16i^2-1}{16i^2-4}$$ I tried doing $\frac{a_{n+1}}{a_n}>1$ to show that the seqeunce is monotonically increasing. (For using the monotone convergence theorem) But I ...
1
vote
1answer
24 views

Is this proof of almost sure convergence correct

I have a sequence of random variables $X_i$ such that $P[X_n = 1 ] = \frac{1}{n}$ and $P[X_n = 0 ] = 1 -\frac{1}{n}$. We can see that this sequence $X_i$ converges in probablity to the sequence $X = ...
1
vote
2answers
46 views

Problem regarding convergency: $\sum_{k=0}^{\infty}\frac{x^k}{k!} \to e^x$

Question. Let $X$ be the space of all polynomials in one variable,with real coefficients. If $p=a_0+a_1x+\dots+a_nx^n\in X$, define $$|p|=|a_0|+|a_1|+\dots+|a_n|,$$ which gives the metric $d(p,q)=|p-q|...
1
vote
1answer
35 views

Convergence of a logarithmic sequence

Is the sequence $\left\{\ln\left((1+\frac1n)^n\right)\right\}_{n=1}^{\infty}$ convergent or divergent?. I tried to solve it by L Hospital's rule and arrived at 0...implying it is convergent..is it? ...
1
vote
1answer
35 views

Why does this tight sequence of random variables also converge in probability?

Definition of convergence in probabilty We say that $X_n$ converges in probability to zero, written $X_n = o_p(1)$, if for every $\epsilon> 0$, $$ P(| X_n | > \epsilon) \rightarrow 0, \...
5
votes
1answer
43 views

Convergence of $\sum\limits_{n=1}^{+\infty}\int_{n^{\alpha}}^{n^{\alpha+1}}\log^2\left(1+\sin1/x\right)\,dx$

Study the convergence of the following series as $\alpha>0$ $$\sum_{n=1}^{+\infty}\int_{n^{\alpha}}^{n^{\alpha+1}}\log^2\left(1+\sin\frac{1}{x}\right)\,dx$$ This is the first time I do an ...
0
votes
0answers
15 views

Can a sequence that converges with order α converge also with an order smaller than α?

Suppose that a sequence {pn} converges to p of order $\alpha$, such that for some $\lambda$ the below equation is established. $$\lim_{n\to \infty} \frac{\lvert p_{n+1} - p \rvert}{\lvert p_n - p \...
0
votes
1answer
25 views

Convergence in probability, mean and almost surely

$X$, $Z_n$, $Y_n$ are independent random variables, where X is integrable, $Y_n$ has Bernoulli distribution $b(1,n^{-2})$ , $Z_n$ has Poisson disribution with parameter $n^2$. I need to check ...
-2
votes
0answers
80 views

Ramanujan's 1/pi Series: Proving that $a_{n+1} < La_n$ (for $n \geq 1$) implies that $a_n< L^{n-1}a_1$ for $n \geq 2$.

I don't know how to go about this question regarding Ramanujan's formula: $$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\sqrt{8}(4n)!(1103+26390n)}{9801(n!)^4396^{4n}}$$ Let $a_n$ denote the nth term ...
2
votes
0answers
35 views

$L^2$-convergence versus locally uniform convergence for a complex power series

Let $M$ be a nice topological space with a nice measure, let's say a Riemannian manifold. Let $B(0,R)$ be a ball in $\mathbb C$ and suppose $a_n : M \to \mathbb C$ are continuous $L^2$ functions such ...
1
vote
1answer
39 views

Convergence in distribution and asymptotic distribution

I am having a problem with proving convergence in distribution (or by law). Consider that the sequence $X_n$ of random variables are IID and that $E[X_n]=0$ and $V[X_n]=1$. Now define the variable $...
5
votes
3answers
59 views

Does $\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$ converge or diverge?

Does this series converge or diverge? If it converges, determine its limit. $$\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$$ So far I said that $\frac{1}{2^n}$ is a geomotric series that converges,...
1
vote
1answer
34 views

Converging Sequences Definition

Q. Explain exactly what it means for $\{a_n\}$ $n\in\mathbb N$ to converge to $L ∈ R.$ I wrote that for $\{a_n\}$ to converge to $L ∈ R$ means that the infinite sequence $\{a_n\}$ has a limit $L$ ...
0
votes
2answers
26 views

Question on convergence of a random sequence after conditioning on a specific event

Consider a random process $\{X_n\}_{n=1}^\infty$ where each random variable is continuous. Assume that the sequence of random variables converges almost surely to $\alpha >0$ i.e., \begin{equation}...
-1
votes
2answers
47 views

$s_n= \frac{1-2+3-4+5-6+7+…+(-2n)}{\sqrt{n^2+1}+\sqrt{n^2-1}}$ [on hold]

What is the limit of this sequence? $s_n= \frac{1-2+3-4+5-6+7+...+(-2n)}{\sqrt{n^2+1}+\sqrt{n^2-1}}$ I need hint. I don't want the solution. Which idea should I use here.
1
vote
2answers
34 views

Epsilon proof of a sequence's limit - algebra issues

I am attempting to prove $\lim_{k\to\infty}\dfrac{k^2}{k^2+2k+2}=1$. However, I am getting tripped up on the algebra. I believe I want to show that there exists some $N\in\mathbb{N}$ such that when $k\...
0
votes
0answers
32 views

A guess on the bound of the product of bounded sequence $\{x_{k}\}$ with $\lim_{t\to\infty}\prod_{k=0}^{t}(1-x_{k})=0$ exponentially.

Let $\{x_{k}\}$ be a bounded sequence, i.e., $|x_{k}|\le T$ for some positive constant $T$. Also, $x_{k}\neq 1$ and $x_{k}\neq 0$. If \begin{align*} \lim_{t\to \infty}\prod_{k=0}^{t}(1-x_{k})=0 \end{...
1
vote
1answer
28 views

Chebyshev's inequality application and convergence - practical example

Let $W_n$ be a random variable with mean $\mu$ and variance $\frac{b^2}{n^{2p}}$, with $p>0$ and $b$ and $\mu$ constants. Show that $$ \lim_{n\to\infty} P(|W_n-\mu| \leq \epsilon) = 1 $$ ...
4
votes
2answers
55 views

A bound for the products of a bounded sequence $\{x_{k}\}$ with $\lim_{t\to \infty}\prod_{k=0}^{t}(1-x_{k})=0$

Let $\{x_{k}\}$ be a bounded sequence with $x_{k}\neq 1$, and if \begin{align*} \lim_{t\to \infty}\prod_{k=0}^{t}(1-x_{k})=0. \end{align*} The question is can we infer \begin{align*} \prod_{k=0}^{t}...