0
votes
2answers
37 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
32 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
30 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
33 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
86 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
21 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
21 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
49 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
39 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
36 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
55 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
35 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
109 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
56 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
27 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
24 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
151 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
104 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} ...
1
vote
0answers
36 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
0
votes
1answer
28 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
2
votes
2answers
71 views

Sum of infinite series $\sum_{n=0}^{\infty}=\frac{n^2}{4n^2-1}t^n$

I have this problem, finding infinite sum of this series: $$\sum_{n=0}^{\infty}\frac{n^2}{4n^2-1}t^n$$ It should be done using derivatives and integrals, like for example: ...
0
votes
1answer
31 views

Suppose $\{x_n\}_{n=1}^{\infty}$ is a bounded divergent sequence. Let $S=$ range$\{x_n\}_{n=1}^{\infty}$. Can $S$ be an infinite set?

So $S$ is the set of all the terms of $\{x_n\}_{n=1}^{\infty}$. I feel like $S$ must be a finite set. I know that $\{x_n\}_{n=1}^{\infty}$ cannot be monotone, since it's bounded and divergent. So if ...
4
votes
1answer
39 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
1
vote
1answer
64 views

Dirichlet's function

How can we see that Dirichlet's function $$D(x):=\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)= \begin{cases} 1 & x\in\mathbb Q\\ 0 & x\notin\mathbb Q\\ ...
0
votes
1answer
41 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
4
votes
2answers
89 views

L. Kronecker's theorem for sequences and series: $\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$

Assume $\sum a_i$ is a convergent series and $b_1,b_2,\dots$ is a divergent monotonically increasing sequence. How can we see that $$\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$$ Attempt: We ...
2
votes
1answer
30 views

Sum of infinite series $\sum_{n=0}^{\infty}(-1)^n\frac{n-1}{n!}t^n$

I have this problem, finding infinite sum of this series $$\sum_{n=0}^{\infty}(-1)^n\frac{n-1}{n!}t^n$$ It should be done using derivatives and integrals, like for example: ...
11
votes
1answer
168 views

The integral on $[0,1]\times[0,1]$

Here I have a problem. $p$ and $q$ are positive numbers. the integral $$\int_0^1\int_0^1 \frac{1}{x^p+y^q}\;dx\;dy< \infty \Longleftrightarrow \frac{1}{p}+\frac{1}{q}>1$$ Here is my try, ...
1
vote
1answer
40 views

$\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$

If $\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) ...
0
votes
0answers
23 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
0
votes
1answer
25 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
0
votes
2answers
73 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...
0
votes
4answers
43 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
1
vote
1answer
22 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
1
vote
1answer
55 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
1
vote
2answers
51 views

Real Analysis Question on convergence

The question is: Show that if the partial sums $s_n$ of the series $\sum\limits_{k=1}^\infty a_k$ satisfy $\vert{s_n}\vert\leq Mn^r$ for some $r<1$, then the series $\sum\limits_{n=1}^\infty ...
0
votes
2answers
27 views

Rearrangement of absolutely convergent series is absolutely convergent??

If $\sum a_n$ is absolutely convergent, is it true that every rearrangement of $\sum a_n$ is also "absolutely" convergent? I know that by the Rearrangement Thm., every rearrangement of $\sum a_n$ is ...
1
vote
1answer
31 views

Question about limit of a product of two real sequences

Let $(a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\subseteq\mathbb R$. Moreover assume that $\lim_{n\to \infty} a_n=c_1\in\mathbb R$ with $c_1\neq 0$ and that $\lim_{n\to \infty} a_nb_n=c$ for some ...
1
vote
1answer
31 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
2
votes
2answers
84 views

convergence of a generalized Riemann integral

Could you please provide me some hints to test the convergence of this integral below? $$\int\limits_{0}^{+\infty}\dfrac{\sin x\cdot \sin 2x}{x^\alpha} \, dx$$ where $\alpha \in \mathbb{R}$
1
vote
1answer
55 views

Convergence of tricky series

I've stumbled upon a particularly unpleasant series, and I can't quite seem to crack it. $\sum\limits_{n=1}^{\infty}\dfrac{\ln(1+nx)}{n^{2}} $ I need to show uniform convergence on any interval of ...
0
votes
1answer
20 views

Determining the domain of convergence of a function

For $$f_n(x)=n\cdot \sin\left(\frac{x}{n}\right)$$ How do I determine for what values of x this series converge? (like $[a,b]$ or $(a,b]$...) thank you very much for any help!
2
votes
2answers
87 views

Prove that $\{a_n\}_{n=1}^{\infty}$ converges to $\frac{x}{2}$.

Let $x$ be any positive real number, and define a sequence $\{a_n\}_{n=1}^{\infty}$ by $$ a_n=\frac{[x]+[2x]+\cdots+[nx]}{n^2} $$ where $[x]$ is the largest integer less than or equal to $x$. ...
0
votes
1answer
21 views

Subsequence to infinity proof

How to prove this: Every not bounded above sequence has subsequence which limit is infinity when n->infinity It's nearly the same what Bolzano–Weierstrass ...
0
votes
1answer
62 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
0
votes
1answer
36 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
0
votes
1answer
33 views

Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...