# Tagged Questions

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### 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 $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
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### 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
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### 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 $B$ ?
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### 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 $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt$$ ...
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### 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 ...
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### 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 $\alpha \in \mathbb{R}$
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### 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): ...
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### 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: $f(x)=\sin{ x}$ ...
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### 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 ...
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### 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 ...
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### 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.
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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): ...
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### 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 ...
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### 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 ...
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### 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 $p>1$.
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### 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 ...
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### 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.
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### Use comparison or limit comparison test to determine whether the series converge [closed]

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
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### 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 ...
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### 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 $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
### 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 ...
### 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 ...