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
21 views

Question about convergence in $H^1_0$

Please how to prove that if $u_n\rightarrow u$ on $H^1_0$ we have that $||u_n||\rightarrow ||u||$ ? Please i need your help Thank you
0
votes
1answer
56 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
1
vote
1answer
34 views

Small question about convergence

I have a small question: if i have that $$\int_0^{+\infty}p(t)|u'_n(t)-u'(t)|^2dt\rightarrow 0$$ is it true that $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt $$ ...
1
vote
1answer
27 views

Question about convergence

I have that $v=v^+-v^-$, $v^+,v^-$ are the positive and the négative part of $v$ and i have this: i dont understand why if $v_n\rightarrow v_0$ in $L^p(\Omega)$ then $v_n^+\rightarrow v_0^+$ in ...
2
votes
2answers
85 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 $\alpha \in \mathbb{R}$
0
votes
2answers
34 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
1
vote
1answer
26 views

Simpson's composite rule rate of convergence.

Hello I have wriiten a program in Matlab that determines an Integral using Simpsons rule and it also determines the rate of convergence. I tried my program on the following examples: $f(x)=\sin{ x}$ ...
1
vote
0answers
55 views

Does $ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right )\right) $ converge? [duplicate]

I am trying to determine whether the $$ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right ) \right) $$ converges or not. I have tried the popular tests, but all ...
3
votes
0answers
34 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...
0
votes
1answer
28 views

Question about link between uniform convergence of sequences and series

How to find functional series $f_n(x)$ that converges uniformly to $f(x)=0$ on some interval I where the series $\sum_{n\ge1}f_n(x)$ converges on this interval but not uniformly ? Thanks.
1
vote
4answers
66 views

A sequence was defined by a equation

The sequence was defined by the equations: $${a}_{1}=a\left(\in R \right),{a}_{n+1}=\frac{2{{a}_{n}}^{3}}{1+{{a}_{n}}^{4}},n\geq 1.$$ show that $\left(a \right)$The given sequence is convergent. ...
0
votes
2answers
80 views

A question about convergence of $b_n=f\left(\frac 1{n^2}\right)$

Suppose $f(x)$ is some function with domain [0,1] and $\sum_{n\ge1}f\left(\frac 1n\right)$ converges, than $a_n=f\left(\frac 1n\right)\to 0$ but does $b_n=f\left(\frac 1{n^2}\right)$ also converge to ...
1
vote
2answers
74 views

Does the series $\sum_{n\ge0}\frac{x^n\sin({nx})}{n!}$ converge uniformly on $\Bbb R$?

The series $$\sum_{n\ge0}\frac{x^n\sin({nx})}{n!}$$ converges uniformly on each closed interval $[a,b]$ by Weierstrass' M-test because ...
3
votes
2answers
46 views

How to show that

Ok, so I've just been banging my head on this one for a bit. This is something I need to prove that: $$\sum_{k=2}^{\infty}(-1)^k\dfrac{\ln|k|^p}{k^q}$$ converges. Obviously, if I can show that the ...
0
votes
2answers
23 views

A question about uniform convergence of $g_n=f\left(\frac xn\right)$

Could you give me some hint how to prove this statement: Suppose $f(x)$ is some function on R. Prove: If $g_n=f\left(\frac xn\right)$ converges uniformly to zero on R than $f(x)=0$ for all x. I ...
1
vote
1answer
15 views

Does absolutely uniform convergence imply uniform convergence ?

Could you please give me some hint how to solve this problem: Prove or provide counter-example: If $\sum|f_n(x)|$ converges uniformly on some interval I then $\sum f_n(x)$ converges uniformly on I. ...
2
votes
2answers
61 views

A question about series convergence at a given point if functional series uniform convergence on interval

Could you give me some hint how to prove this statement: If $f_n(x)$ is continuous sequence on [0,1] and the series $\sum_{n\ge1}f_n(x)$ converges uniformly on [0,1) then the series ...
1
vote
1answer
52 views

Investigating the convergence of a series using the comparison limit test

Actually not sure how to approach this... but I may be missing something: Replacing the sequence: $x_{n}=1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}-2\sqrt{n},\,\,\,\, n=1,2,....$ By the ...
1
vote
1answer
19 views

What is the probability limit and limit distribution of the estimators given that$ X_i$ are iid

This is more of a practice question but I'm not sure how to really proceed. Say that $E(X)=0$ and $Var(X)=\sigma^2$. Firstly I am required to find the probability limit of the estimator as $n$ go to ...
1
vote
2answers
58 views

Check if $\int_1^{\infty\:}\left(e^{-\sqrt{x}}\right)dx$ converges using Convergence Test

I could use some help with an homework question: Using the convergence test, check if the following integral function converges or diverges (no need to calculate the limit itself): ...
3
votes
1answer
71 views

Does $\sum_{n\ge1}\frac1n\sin\left(\frac xn\right)$ converges uniformly?

On each bounded interval $[a,b]$ : $\left|\frac1n \sin\left(\frac xn\right)\right|\le \frac{\max\{|a|,|b|\}}{n^2}$, the series $\sum_{n\ge 1}\frac {\max\{|a|,|b|\}}{n^2}$ converges, therefore ...
2
votes
3answers
57 views

Proof a sequence converges to a limit

For a sequence $$a_n = \frac{\sin(n)+2}{4n^2-28}$$ How would you use the definition of a limit of a sequence to prove $a_n$ converges to $0$ I am really stuck with how this definition works, I ...
4
votes
1answer
44 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
0
votes
1answer
24 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
1
vote
2answers
56 views

A question about convergence interval of power series

Could you give me some hint how to solve this problem: Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$. I managed to prove that $a_n$ is decreasing ...
0
votes
1answer
21 views

Convergence of parametric series.

Could you give me some hint how to decide about convergence of series $\sum_{n\ge1}\left(\left(-1\right)^n+\alpha^3\right)\left(\sqrt{n+1}-\sqrt{n}\right)$, where $0<\alpha\le1$. It obvious that ...
2
votes
2answers
46 views

Does the series $\sum_{n\ge1}\frac{(-1)^n}{\left(n\ln\frac{n+1}n\right)^n}$ converge?

It seems to me that the general term of this series is not tends to zero : $\left(n\ln\left(1+\frac1n\right)\right)^n\sim n^n\frac1{n^n}=1$ so $\frac 1{\left(n\ln\frac{n+1}n\right)^n}\ge1$. Am I ...
1
vote
1answer
62 views

Does the series $\sum_{n\ge1}\frac{ln\left(\frac{n+1}n\right)}{\sqrt n}$ converge?

Could you please give me some hint how to decide about convergence of the series $\sum_{n\ge1}\frac{ln\left(\frac{n+1}n\right)}{\sqrt n}$ ? I tried using comparison test: ...
1
vote
1answer
31 views

Power Iteration method for eigenvalues - Show the error is bound

Let $A \in $Sym$_{n}(\mathbb R)$ with eigenvalues $\lambda_i$ such that $|\lambda_1| > |\lambda_2| \geq |\lambda_3 |\geq ... \geq |\lambda_n|$ We define the following process as "Power Iteration": ...
2
votes
0answers
34 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
2
votes
1answer
72 views

Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
0
votes
3answers
111 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
1
vote
0answers
56 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
0
votes
1answer
39 views

Pointwise convergence and uniform convergence of a sequence of functions

Let $\{f_{n}\}_{n\geq 1}$ be a sequence of function given by $f_{n}(x)=\frac{1}{x}+\frac{1}{n}$. Does $f$ converge pointwise on $\mathbb{R}\setminus\{0\}$? Does $f$ converge uniformly ...
0
votes
1answer
47 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
0
votes
2answers
51 views

Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence but the convergence is not uniform on $[0,1]$

Let $$ f_n(x) := \begin{cases} 1 &\text{for $x$ in } \left(0, \frac{1}{n}\right)\\ 0 &\text{$x$ elsewhere in } [0,1] \end{cases}. $$ Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence ...
0
votes
1answer
32 views

Pointwise limit,$f$, of the sequence is not bounded

Question: Let $f_n(x) := \frac{nx}{1+nx^2}$ for $x \in A := [0, \infty)$. Show that each $f_n $is bounded on $A$, but the point-wise limit of $f$ of the sequence is not bounded on $A$. Does $(f_n)$ ...
2
votes
1answer
28 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
1
vote
1answer
43 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
1
vote
1answer
58 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
0
votes
2answers
30 views

A question about series ratio test

Could you please give me some hint how to deal with this question: Suppose $\left|\frac {a_{n+1}}{a_n}\right|\le c_n$ for each n and $c_n<1$. May we conclude that $\left|\frac ...
0
votes
1answer
48 views

Convergence parameter: Find the value of $p>0$ for which the series converge

For the sum for $k=2$ to infinity: $$\frac{\ln k}{k^p}\ $$ The textbook says the answer is $p>1$.
0
votes
1answer
33 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
2
votes
1answer
37 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
0
votes
1answer
29 views

Use comparison or limit comparison test to determine whether the series converge [closed]

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
2
votes
2answers
68 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
1
vote
1answer
57 views

Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
1
vote
1answer
41 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
1
vote
2answers
85 views

The limit of $\ln(n) - \ln(n^2 + 1)$ as $n\to\infty$

As $n\to\infty$, what is the limit of $\ln(n) - \ln(n^2 + 1)$ Using properties of logs and limits, I ended up with: $$ \ln \left(\lim \left(\frac{n}{n^2 + 1}\right)\right) $$ where lim is the ...