0
votes
1answer
37 views

$\{X_n\}$ is a bounded, divergent, infinite sequence of real numbers, prove [on hold]

prove (A) $\{X_n\}$ contains convergent subsequences with different limits. (B) $\{Y_n = \min_{k\le n} X_k\}$ is convergent. not sure if (A) is correct.
1
vote
0answers
24 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
10
votes
4answers
144 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
1
vote
2answers
58 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
3
votes
1answer
21 views

Convergence of $\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$

For which $\alpha$ (depending on $n$) does $$\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$$ converge? Examples: For $n=1$ the series turns out to be $$\sum_{i=1}^\infty ...
1
vote
1answer
32 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
2
votes
1answer
45 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
2
votes
1answer
28 views

Show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$.

This should be a relatively easy question, but I can't seem to figure it out. I want to show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$. As ...
0
votes
2answers
42 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
3
votes
2answers
71 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
0
votes
1answer
31 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
1
vote
1answer
32 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
2
votes
2answers
61 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
1
vote
0answers
45 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
1
vote
2answers
64 views

Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $?

I have being trying to solve this convergence but with no success. Using the ratio test I have reached here: $$ a_n = \left( \frac{n}{n+1} \right)^{n^2} $$ And $$ \frac{1}{a_n} = \left( ...
2
votes
2answers
57 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
2
votes
0answers
36 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
0
votes
1answer
51 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
-4
votes
2answers
100 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
2
votes
1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
4
votes
3answers
125 views

$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
50 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
0
votes
2answers
52 views

Finding the convergence radius of $\dfrac{(n!)^k\cdot x^n}{(kn)!}$

If K is a integer positive, find the convergence radius of the series $$\sum\limits_{n=0}^{\infty} \dfrac{(n!)^k\cdot x^n}{(kn)!}$$ Any initial idea?
5
votes
1answer
43 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
2
votes
1answer
40 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
3
votes
1answer
32 views

Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
0
votes
0answers
35 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
1
vote
3answers
53 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
0
votes
2answers
39 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
0
votes
0answers
33 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
1
vote
1answer
31 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
4
votes
1answer
34 views

A question about the convergence of series.

If$\hspace{0.3cm}$ $\sum a_n$ and $\hspace{0.3cm}$$\sum b_n$$\hspace{0.3cm}$ are convergent then which of the following is true? $1.$$\hspace{0.3cm}$$a_{n+1}<a_n$$\hspace{0.3cm}$ $\forall n$ ...
5
votes
0answers
93 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
0
votes
1answer
25 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
3
votes
2answers
33 views

What is the range of convergence of $\sum_{k=0}^\infty (k \cdot x \exp(-x))^k\cdot {1 \over k!} \cdot {1\over k+1}$?

I have the following series as an expression which occurs as a limit of a quotient of polynomials in $e$ and $x$ which I've expanded by polynomial long division into a series: $$f(x) = ...
1
vote
1answer
23 views

Sequence problem dealing with continuity and convergence.

I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...
2
votes
2answers
50 views

Convergence of integrals but $\int_a^b|f_n(x)-f(x)|dx$ does not converge to $0$

Can I have an example of a sequence of continuous functions $(f_n)_n$ and a continuous function $f:[a,b]\to \mathbb{R}$ such that $$\int_a^bf_n(x)dx\to\int_a^bf(x)dx,$$ when $n\to+\infty$, but ...
1
vote
1answer
42 views

Interval of convergence

Find the interval of convergence of $\sum_{n=1}^{\infty }\frac {n^{2n}}{(2n)!}x^{n}$ I use ratio test and i found $\lim_{n\to\infty}|\frac{a_{n+1}}{a_{n}}|<1$ iff ...
0
votes
0answers
31 views

A variant of the Riemann Integral

This question is related to this one. Let $S_k$, $k\in\mathbb{N}$ be a sequence of finite sets where $S_k\subset S_{k+1}\subset[0,1]$. Fixed $s$ in $S_k$, let $s'$ denote the predecessor of $s$ in ...
0
votes
1answer
41 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
1
vote
0answers
63 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
0
votes
1answer
40 views

How to deal with discontinuous points when proving that step functions are dense in $PC[a,b]$

This question is a follow-up to my previous question: How does one prove that a space is dense in another under some norm? I figured out a way to solve (part of) the exercise. Given some function ...
1
vote
2answers
113 views

Why does the sequence ${1/n}$ not converge in the positive reals?

I'm reading Baby Rudin at the moment and it claims something remarkable. Consider the sequence $$ x_n=\frac{1}{n}.$$ The book claims that this converges to zero in the reals: $$\lim_{n\to\infty} ...
0
votes
2answers
57 views

Convergence Proof Problem $\epsilon, n_0$ proof

I need help proving that for the following sequence it will converge to the given limit $p $using an $\epsilon$ $n_0$ argument (i.e given $\epsilon>0$, determine $n_0$ such that $|p_n - p| < ...
3
votes
1answer
32 views

How to prove the sequence $\{b_n\}$ with $0 \leq b_{n+1} \leq b_n + a_n$ converges, where $a_n \geq 0$ converges to $0$.

Let $\{a_n\}$ be a sequence of non-negative real numbers that converges to zero. Suppose the sequence $\{b_n\}$ of non-negative real numbers satisfies \begin{equation} 0 \leq b_{n+1} \leq b_n + a_n ...
0
votes
1answer
14 views

Does this function go to zero faster than the norm of its argument?

Assume $f:\mathbb R^2\to\mathbb R$ is such that for all $\varepsilon>0$ exists $\delta>0$ such that, whenever $||x||<\delta$, also $||f(x)||<\varepsilon^2$. Can we see that $f$ is ...
0
votes
1answer
26 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
4
votes
4answers
153 views

Convergence of $\left(1+\frac{1}{n}\right)^n$, given that $\left(1+\frac{1}{2^n}\right)^{2^n}$ converges

I know how to prove that the sequence $$\left(1+\frac{1}{2^n}\right)^{2^n}$$ has a limit. Can I use this knowledge to quickly get the fact that $$\left(1+\frac{1}{n}\right)^n$$ also has a limit?
6
votes
1answer
106 views

The sequences $x_n$ and $y_n$ converges

Let $\quad2{x}_{n+1}=1+{y}^{2}_{n},\quad 2{y}_{n+1}=2{x}_{n}-{x}^2_{n},\quad n\in\mathbb{N};\quad 0\leq {y}_{0}\leq \frac{1}{2}\leq {x}_{0}\leq 2.$ Prove that the sequences $ \begin{Bmatrix} ...