Convergence of sequences and different modes of convergence.

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Is there a better solution (?)

There are two sequences $ a_n $ and $ b_n $, which fulfill 3 certain properties: The series of one of them converges, while the other diverges. $ \frac {a_n} {b_n} $ = 1, if n tends to infinity. ...
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3answers
35 views

Prove convergence for p>1

Prove the series is convergent for p>1: $\sum _{k=1}^{\infty } \frac{1}{(n)(ln(n)^p)}$ The ratio test is inconclusive. I assume a comparison test is involved but am not sure how to make that happen. ...
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20 views

$L^2$ limit of Gaussian random variables

Let $X_n$ be a sequence of Gaussian random variables defined on the same probability space. The statement is that if $X_n$ converges to some random variable $X$ in $L^2$-sense, then $X$ is also ...
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1answer
29 views

Power series with $x^{4n}$

I'm new to this Forum. I do not find an approach to solve the following problem (from the book "Herbert Wallner, Aufgabensammlung Mathematik Band 1", so this is not a homework question): For which ...
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0answers
46 views

What's the general technique to show a sequence converges?

After "guessing" what the limit of a particular sequence is, what's the general process to prove that this sequence indeed converges to it? (using the definition) (The definition says that a sequence ...
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2answers
60 views

Prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent

How do I prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent for every x in real numbers except for $x=0$? I tried using the ratio test, but it ...
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Get stuck with some statements of convergence rate of the iteration method from “Iterative methods for sparse linear systems (2nd edition) ”

Here are the statements I get from the book and the two highlight parts are what I can not understand well. The questions are: Why we can conclude that $\rho=\rho(G)$ from"the above analysis"? ...
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38 views

Find the radius of convergence, R, of the series.

Find the interval $I$ of convergence of the series $$\sum_{n=1}^{\infty} 9 (-1)^n n x^n.$$ (Enter your answer using interval notation.) I'm stuck with this problem and could use a ton ...
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1answer
27 views

Price of a commodity converges to a limiting price

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k=a+bp_k$ and the supply depends on ...
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2answers
23 views

Convergence and uniform convergence of $f_n(x)$

The given sequence of functions is defined as $f_n(x) = \frac{x^n}{n+x^n}$ for $x\ge 0$ and $n = 1,2,\ldots$; let $f = \begin{cases} 0 &:0\le x \le 1\\ 1 &:x>1 \end{cases} $ My hypothesis ...
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4answers
107 views

Establish the convergence and find the limits of the following sequence

$a_n = \left(1+\dfrac{1}{n}\right)^{n+1}$ I know that the answer is supposed to be $e$ but I am unsure how to reach that answer. I am so lost where to even begin with this
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1answer
29 views

Did I correctly verify the convergence of this series?

I want to find if the following series is convergent. $$\sum_{n=1}^\infty \frac{(1+\frac{1}{n})^nn^2-7n}{n^3+3n^2+1} $$ I use the asymptotic criterion for series convergence. $$ ...
2
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2answers
30 views

Uniform Convergence of $1/n\sin(n^​2 x)$

For $n\in\mathbb{N}$, define $f_n:(0,1)\to\mathbb{R}$ by $f_n(x)=1/n\sin(n​2 x)$. Does $f_n$ converge uniformly on $(0,1)$? 1/nsin(n squared x) Can someone show me how this converges uniformly as I ...
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1answer
42 views

convolution with $C^{\infty}$ produces $C^{\infty}$

Problem: So I have the following function in ...
2
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0answers
61 views

Basic Fourier analysis explanation needed wrt a function $f$ and a finite Borel measure $\mu$

An extract from Chapter 12 of Matilla's Geometry of Sets and Measure on Euclidean Spaces I do not believe that formulas (12.1-12.3) are easily seen to be valid. I do not understand what ...
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1answer
25 views

Convergence of integral of multiplication of two positive functions

I have two functions $f, g:\mathbb{R}\rightarrow \mathbb{R}_{\ge0}$, that are continuous. I know that $\int\limits_{-\infty}^\infty f(s) \, ds=C_1<\infty$, and $g(s)\le C_2$, with $C_1> 0$ and ...
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51 views

Give an example of convergent sum

I do it like that but my teacher told me that an^2 diverge since n^1/2 by p-series diverge.Can someone help me
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1answer
32 views

Does convergence in distribution of discrete random variables with same finite support imply convergence in probability?

Let ${X_m}\mathop \to \limits^D Y$ and ${\text{supp}}\left( {{X_m}} \right) = {\text{supp}}\left( Y \right) = \left\{ {0,1, \ldots ,n} \right\},\forall m \in \mathbb{N}$. Does ${X_m}\mathop \to ...
2
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0answers
21 views

Convergence of a kinda bump function to Dirac delta

I'm a bit stuck with the following question. I'd like to know if it is true that $$f_\epsilon(\theta):=\frac{M_\epsilon(\theta)}{\int_{-\pi/2}^{\pi/2}M_\epsilon(\alpha)d\alpha}$$ where ...
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20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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39 views

Weak convergence to a constant implies convergence in probability

If on some probability space the random variables $X_1, \dots, X_n$ with distributions $\mu_n$ convergence weakly to the constant random variable $c \in \mathbb{R}$, i.e. $ \int f \, d \mu_n \to ...
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1answer
20 views

Regarding pointwise convergence

I came across the following function that is supposed to converge pointwise to the zero function. But I cannot understand how. $$ f_{n}(t) = \begin{cases} n^2t,&0\le t\le\frac1n\\ ...
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2answers
66 views

Is the series $\sum e^{an^2}(1-\frac{a}{n})^{n^3}$ convergent?

I stumbled upon this sum: $$\sum e^{an^2} (1-\frac{a}{n})^{n^3}$$ Wolfram Alpha tells me it is convergent but I can't find a convenient proof to use. The $n$-th root test seems to be useful, since ...
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31 views

Limits in matrix norm if convergence of integration is guaranteed

the answer of this question probably is very obvious, but I want to make sure this is correct. I have a function $F: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ that is continuous and ...
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3answers
55 views

Prove that $(y_n)$ converges

Prove that $(y_n) = \frac1n\sin({n\pi\over3})$ converges Now I know my RTP: ($\forall\epsilon\gt0)(\exists k \in N)(\forall n \gt k) \\ |(y_n)-c| \lt \epsilon $ but from there i get stuck.
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2answers
35 views

Convergence of an infinite series?

I feel the coefficient Cn has to be zero in order for the original series to converge, as the power series of 4^n will diverge as n - > ∞. Are there any other ways for this series to converge, and ...
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1 views

Asymptotic distribution of ratio / multiplication of two variables

Suppose $\rightarrow_D $ denotes convergence in distribution. If we know $$ f_1 \rightarrow_D W_1 $$ $$ f_2 \rightarrow_D W_2 $$ Can we say something about the convergence of $$ f_1 f_2 ...
2
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1answer
75 views

If $ u_{n+2}\leq \frac{1}{2}(u_n+u_{n+1})$ then $u_n$ converges.

Let $u_n\in \Bbb{R}^\Bbb{N}$ be a positive sequence such that $ u_{n+2}\leq \frac{1}{2}(u_n+u_{n+1}).$ Show that $u_n$ is convergent. Usually I don't post question where I don't have anything ...
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1answer
17 views

Prove an infinite sequence converges given another convergent sequence.

I have some ideas about this question but I'm not sure it's right. Heres the question: Suppose $(x_n)_{ n\ge1}$ is a sequence of real numbers converging to $x$. Define a sequence $(y_n)_{ n\ge1}$ ...
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1answer
17 views

Proving convergence or divergence

I have to show the following diverges: $$\int_0^1 \frac{1}{x^{1/3} -x^{4/3}} dx$$ I am meant to do this without evaluating the integral. I know that I have to split it into: $$\int_0^{0.5} ...
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2answers
42 views

For which of the following choice of $a_k$ is $\sum a_k$ convergent?

For which of the following choice of $a_k$ is $\sum a_k$ convergent? i)$\frac {sinh(k)}{2^k}$ ii)$(1-\frac{1}{k})^{k^2}$ Honestly, I have no idea. Usually, when I see $sin$ or $cos$, I use ...
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4answers
94 views

Investigate convergence of the following series

Investigate the corvergence of the following series? $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+\frac{1}{13}+\frac{1}{15}-\frac{1}{8}+ \ldots$ ...
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45 views

Prove that the given sequence converges to zero [duplicate]

$\sqrt{n+3} - \sqrt{n}$ I have multiplied the top and bottom by $$\frac{\sqrt{n+3}+ \sqrt{n}}{\sqrt{n+3}+ \sqrt{n}}$$ and obtained $$\frac{3}{\sqrt{n+3}+\sqrt{n}}$$ How do you or can you simplify ...
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Function of probability measures representation

Let $\mu$ be a in $\mathbb M(\mathbb R)$, the space of probability measure on $\mathbb R$. Let $F$ be in $C_b (\mathbb M(\mathbb R), \mathbb R)$, the space of continuous bounded function on $\mathbb ...
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1answer
66 views

Prove that the sequence converges to zero [closed]

$$a_n=\sqrt{n+3} - \sqrt n$$ Can someone please give me a detailed way of how to prove that this sequence converges to zero?
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3answers
98 views

A series involving $\prod_1^n k^k$

Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent? My attempt was to use the comparison test, but I'm stuck at finding the behaviour of ...
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1answer
77 views

Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges

How to prove $$\sum_{k=1}^{\infty} \frac{\sin(kx)}{k}$$ converges without using integral test?
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2answers
33 views

Do these converge?

For the following choices for $a_k$, use the indicated test to show whether $\sum a_k$ converges or diverges $\frac{1}{k^{1/k}k}$ (Comparison Test, limit form) $\binom{2k}{k}^{-1}(4-10^{-23})^k$ ...
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1answer
45 views

Weierstrass $M$-Test Example

Does the power series about $0$ for $\frac{1}{1+x^2}$ converge uniformly for $x \in (0,1)$? I am trying to use the Weierstrass $M$-Test. This is what I have so far: $\frac{1}{x^2+1} = ...
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2answers
43 views

Prove that series converge [duplicate]

Show if series $\displaystyle \sum_{n=1}^{\infty}a_n$ has a positive terms and converge then the series $\displaystyle \sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{n}$ converge too. I don't have idea what ...
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2answers
50 views

prove $\sum \frac{x^n}{1+x^{2n}}$ converges

$\sum \frac{x^n}{1+x^{2n}}$ where $x>=0$ I can show that this diverges when $x=1$ and it definitely converges when $x=0$. However I am having trouble showing it converges otherwise(when $x$ is not ...
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40 views

Convergence of a two distinct series [closed]

Investigate convergence of a series: $-\frac{1}{2}+\frac{1}{6}-\frac{1}{10}+\frac{1}{14}-\frac{1}{18}+\ldots$ and $-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\ldots$ ...
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1answer
43 views

$How to determine if this series is convergent?

Let the sequence $\{a_n\}$ be defined as follows: $$a_n \colon= \begin{cases} \frac{1}{n^2} \mbox{ if $n$ is not the square of any positive integer}; \\ \frac{1}{\sqrt{n}} \mbox{ if $n$ is the ...
4
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2answers
48 views

Show that $nu_n$ converges to $1$.

$\Bbb{K}=\Bbb{R}$ or $\Bbb{C}$ Let $(u_n)\in\Bbb{K}^{\Bbb{N}}$ be a sequence such that $(n(u_n+u_{2n}))_{n\in \Bbb{N}}$ converges to $\frac{3}{2}$ and $u_n\rightarrow 0$. It is asking to ...
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2answers
18 views

Pointwise/Uniform Convergence of a function

Let $f_n(x)$=$nx ,\space \space x\in[0,\dfrac{1}{n}]\brace 0, \space\space x\in(\dfrac{1}{n},1]$. Find the pointwise limit of $(f_n)$ and check if $(f_n)$ is uniformly convergent.
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1answer
40 views

Proving a certain limit for uniformly integrable random variables

There is an interesting problem that has been resisting my efforts for a while. Assume that $\{X_n: n = 1, 2, \ldots\} $ is a sequence of uniformly integrable random variables. I would like to show ...
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1answer
48 views

Deduce that $e$ is irrational from the following inequality…

Deduce that $e$ is irrational from the following inequality.... $0 < e-\sum\limits_{k=0}^{n}\frac{1}{k!}<\frac{1}{n!n}$ where $n\geq1$ Fairly straightforward to show that $0 < ...
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2answers
30 views

Prove convergence by considering the partial sums

Let $p$ be a non-zero natural number. Prove by considering the partial sums that $\sum \frac{1}{k(k+p)}$ converges. What is $\sum\limits_{k=1}^{\infty} \frac{1}{k(k+p)}$ No idea. Obviously, it ...
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1answer
34 views

Demonstrate the existence of the following limit

Prove that for $m \geq n \geq 1$ that $|a_m-a_n| \leq n^{-1}$ and deduce that $(a_n)$ converges. For $n\in \mathbb{N}$, denote $$a_n=\int\limits_1^n\frac{\cos(x)}{x^2}dx.$$ By integration by parts, ...
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1answer
61 views

Newton's method convergence implementation

How can I solwe this problem: Experimentaly examine convergence Newton's method for conformation: \begin{align} 2x^3-y^2-1=0 \\ xy^3-y-4=0 \end{align} for various loaded inputs with start points ...