Convergence of sequences and different modes of convergence.

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29 views

Checking if $(x^{2n}-1.5x^n+\frac12)^\infty_{n=1}$ converges uniformly on$ [\frac12,1]$ and on $[0,\frac12]$

Checking if $\left(x^{2n}-1.5x^n+\frac12\right)^\infty_{n=1}$ on $ [\frac12,1]$ and on $[0,\frac12]$. What tests/techniques do I use to prove uniform convergence?
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2answers
37 views

Convergence of an improper integral $I=\int_0^\infty \frac{\sin{x}-x\cos{x}}{x^\alpha}dx$

Problem: Analyze convergence of an improper integral $I=\int_0^\infty \frac{\sin{x}-x\cos{x}}{x^\alpha}dx$. My work: Problematic points are $0$ and $\infty$. Therefore, we will write integral as a ...
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0answers
115 views

Multivariate convergence in distribution

Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$. Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$. Prove that convergence in Distribution of $$ \sqrt n \left(\...
3
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1answer
42 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...
2
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1answer
47 views

Uniform integrability in central limit theorem

Suppose $X_1,X_2,\ldots$ are i.i.d. with $P(X_1=+1) = P(X_1=-1) = \frac 12.$ We know that $n^{-1/2}\sum_{i=1}^n X_i \stackrel{d}{\to} Z$ where $Z\sim\mathcal{N}(0,1).$ How steeply can a continuous ...
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3answers
32 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...
2
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1answer
38 views

Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
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0answers
17 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: approach 1: Assuming $$\lim E[\exp\{aS_n - bn\}] = E[\lim \exp\{aS_n - bn\}]$$ I can't seem to be able to prove $$\lim E[\exp\{aS_n - bn\}] = 0$$ with just $b &...
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2answers
96 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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5answers
68 views

how to determine if this series converges?

I was trying to find out if the series: $$\sum^{\infty}_{n=1}n^3e^{-n} $$ converges. I tried applying the Cauchy test, $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{n^3e^{-n}}=\...
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1answer
18 views

$((n-1)^{0.5})/(((n+1)^2)-1)$ Is the sum convergent?, why or why not? [on hold]

$$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks.
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1answer
21 views

Sufficiently proving a sum converges

This might be a bit basic, but I'm pretty sure I'm wrong about this so I'd appreciate at least a confirmation that I'm wrong. The question is: Prove or disprove: If $\sum_{n=1}^{\infty} ...
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0answers
14 views

Radius of convergence for the following power series using the ratio test

I am slightly unsure about how to do the following question relating to the radius convergence (using specifically the ratio test). The power series is as follows: $$\sum_{n=1}^{\infty}\frac{(2x+1)^n}...
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2answers
285 views

Limit of $\int_0^1\frac1x B_{2n+1}\left(\left\{\frac1x\right\}\right)dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
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1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
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2answers
38 views

For which $ \alpha >0 $ the integral $\int _0^3\:\frac{x}{\left(9-x^2\right)^{\alpha }}dx$ converges?

Good evening to everyone. I have a problem and I don't know where to strat from. I have to find for which $ \alpha >0 $ the integral $$\int _0^3\:\frac{x}{\left(9-x^2\right)^{\alpha }}dx$$ ...
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4answers
51 views

For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment?

For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment? To examine convergence I usually use rules for convergence, but what ...
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1answer
19 views

Proof radius of convergence for zero and infinite (power series)

I deleted my last question because there was a huge mistake inside. Given: $R$ is the radius of convergence of $\sum_{n=0}^{\infty} a_{n}x^{n}$, also suppose that $\lim_{n\rightarrow \infty} \left | \...
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2answers
59 views

How to determine convergence of a rooted parameter?

Ok, so this one stumped me completely. Determine for which values of $\alpha$ the following sum converges: $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{4+n^\alpha}}$$ My gut instinct was that for any $...
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1answer
47 views

Show that shifting a sequence by $M$ positions does not change its covergence or limit. [on hold]

Suppose $(a_n)$ is a sequence, and $M$ is a fixed positive integer. We define a new sequence $(b_n)$ by $b_n = a_{M+n}$. (so the new sequence is the old one ‘shifted’ by $M$ terms.) Exercise 1. ...
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0answers
31 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
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1answer
65 views

Rearrange the series $ \sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ to converge to $1$.

I have studied the Riemann's theorem about rearrangement of conditionally convergent series. Also I have seen other rearrangements of the given series on this site that converge to different sums $\...
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3answers
115 views

Does $\int_0^{1/2} \frac{1}{x\ln x}dx$ converge?

I tried this: $$ \begin{align*} \ln x &= t \\ \frac{1}{x} dx &= dt \\ \lim_{x \to 0^+} \ln x &= -\infty \end{align*} $$ So now we have $$ \int_{-\infty}^{\ln(1/2)} \frac1t dt $$ which ...
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15 views

Vector elements converging to the same value - a proof by contradiction

Note: I'm going to simplify the proposition and proof in this question a bit to avoid a large number of definitions and theorems - hopefully I don't remove anything vital. I'm afraid the material here ...
2
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3answers
26 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
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2answers
88 views

A convergent series: $\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$

I would like to find the value of: $$\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$$ I could only see that the ratio of two consecutive terms is $\dfrac{1}{27\cos(2\theta)}$.
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2answers
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Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
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0answers
17 views

Given a certain half-life, which expression yields the greatest amount of chemical left?

Say I have a chemical with a half-life of 12 hours. I can dissolve this chemical into water, and after 12 hours, exactly half of the chemical is depleted. Now, in this thought experiment, there are ...
4
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1answer
34 views

Find the radius of convergence of this power series: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
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1answer
27 views

Problem involving sequence of random variables on probability space [on hold]

How do I construct (and prove that) an example of a sequence of random variables $\{X_n\}_{n\, \ge\, 1}$, on an appropriate probability space, for which $X_n$ converges to $0$ in $L^r$ for all $r > ...
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1answer
33 views

does this converge? [on hold]

If I have $$X_n=\begin{cases}x_n & p_n\\ 0 & 1-p_n \end{cases}$$ and I know that $x_n$ converges to $0$ as $n$ tends to $0$, can I say that $X_n$ converges to $0$ almost sure?
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2answers
36 views

How to determine the convergence of the following series?

Good evening to everyone! I have the following series $$ \sum _{n=1}^{\infty }\left(-1\right)^{n }\left|\alpha -1\right|^n\frac{n!}{\left(n+1\right)!-n!+1} $$. I don't know from where to start to ...
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2answers
34 views

Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
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0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
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1answer
38 views

Does this converge?

If I have $$X_i=\begin{cases}2\quad p=\frac{1}{3}\\ \frac{1}{2}\quad p=\frac{2}{3} \end{cases}$$ random variables with the same distribution. How can I compute the limit almost sure as $n\to\infty$ ...
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1answer
81 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...
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1answer
41 views

Convergence of $\sum_{n=0}^\infty n^{1/n}-1$ and $\sum_{n=0}^\infty (1/n!)^{1/n}$

$$\sum_{n=0}^\infty n^{1/n}-1$$ $$\sum_{n=0}^\infty (1/n!)^{1/n}$$ Hi. I am working on calculus now. While studying convergence test part, I ran into those problems... Wolfram alpha says they both ...
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1answer
93 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
6
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4answers
372 views

Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$

Please help me to prove tha $$\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$$ converges and its limit is $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\...
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The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows \begin{equation}\label{cp26} F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...
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0answers
17 views

problem in convergence and uniform convergence of sequence of functions?

Let $f: \mathbb{R} \to [0, \infty)$ be a non- negative real-values continuous function. Let $$ \phi_n(x) = \begin{cases} n, \ \quad if \ f(x)\geq n\\ 0, \ \quad if \ f(x) < n\end{...
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1answer
43 views

Interval of convergence for series with complex numbers

I'm trying to find interval of convergence of this series: $$\sum_{n=1}^{\infty} \frac{7^n(z+2i)^n}{4^n+3^ni}$$ and I should draw a plot which represents the answer, this is what I've got so far: ...
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3answers
81 views

Convergence problem $\sum \left(1-n\sin\left(\frac{1}{n}\right)\right)$ [closed]

I have to check convergence of: $$\sum_{n=1}^\infty\left(1-n\sin\left(\frac{1}{n}\right)\right).$$ I have no idea but I only check that $\lim \ n\left(1-n\sin\left(\frac{1}{n}\right)\right)=0$.
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0answers
34 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
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2answers
39 views

An example of a power series that has a radius of convergence of 3

The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if ...
2
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3answers
77 views

Convergence of $\sum a^{1/x_n}$ for $a$ in $(0,1)$ and $\sum x_n$ a positive convergent series

Let $\sum x_n$ be a convergent series of positive real numbers and $0<a<1 $, then is the series $\sum a^{1/{x_n}}$ convergent ? I have only figured out that $\lim a^{1/{x_n}}=0$.
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0answers
19 views

Showing $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ is a Hilbert space

Let $I$ be an open interval in $\mathbb{R}$. We define $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ with the scalar product of the Sobolev space $H^1(I)$, i.e. $(u,v)=(u,v)_{L^2(I)}+(u',v')_{L^...
0
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1answer
24 views

Radius of convergence of power series of $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ at $0.$

The function $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ has a power series expansion in a neighborhood of the origin. What is its radius of convergence. I believe I have to use the ratio test and show that ...
1
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1answer
23 views

Show that MLE estimator convergences in probability to actual parameter

For iid stochastic variables $X_1, ..., X_n$ whose distribution is defined by 2 parameters, I have found the MLE estimators. They are $\hat{\mu} = \sum x_i/n$, and $\hat{\lambda}$ given by $$ \frac{...