prove that $f_n(x) = frac{n^2x}{1 + n^2x^2log n}$ converge but not uniform












2












$begingroup$



I have to prove that the sequence of functions
$$
f_n(x) = frac{n^2x}{1 + n^2x^2log n},
$$ converges.




So I think that I can just show that $lim_{n to infty} (f_n) = 0$ pointwise. But to show that the sequence does not converge uniformly, I need to show that $f_n$ has a point of maximum $x = frac{1}{n sqrt{log n}}$ and at this point, the limit is equal to infinity. Is that correct?



And finally I need to show that $forall alpha > 0$ in the interval $mathbb{R} setminus (-alpha,+alpha)$, the sequence $f_n$ converges uniformly, but I don't have ideas how to do it.






real-analysis sequences-and-series analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 21:39










  • $begingroup$
    Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
    $endgroup$
    – Matheus Fachini
    Dec 15 '18 at 21:45






  • 1




    $begingroup$
    Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 22:23
















2












$begingroup$



I have to prove that the sequence of functions
$$
f_n(x) = frac{n^2x}{1 + n^2x^2log n},
$$ converges.




So I think that I can just show that $lim_{n to infty} (f_n) = 0$ pointwise. But to show that the sequence does not converge uniformly, I need to show that $f_n$ has a point of maximum $x = frac{1}{n sqrt{log n}}$ and at this point, the limit is equal to infinity. Is that correct?



And finally I need to show that $forall alpha > 0$ in the interval $mathbb{R} setminus (-alpha,+alpha)$, the sequence $f_n$ converges uniformly, but I don't have ideas how to do it.






real-analysis sequences-and-series analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 21:39










  • $begingroup$
    Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
    $endgroup$
    – Matheus Fachini
    Dec 15 '18 at 21:45






  • 1




    $begingroup$
    Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 22:23














2












2








2


1



$begingroup$



I have to prove that the sequence of functions
$$
f_n(x) = frac{n^2x}{1 + n^2x^2log n},
$$ converges.




So I think that I can just show that $lim_{n to infty} (f_n) = 0$ pointwise. But to show that the sequence does not converge uniformly, I need to show that $f_n$ has a point of maximum $x = frac{1}{n sqrt{log n}}$ and at this point, the limit is equal to infinity. Is that correct?



And finally I need to show that $forall alpha > 0$ in the interval $mathbb{R} setminus (-alpha,+alpha)$, the sequence $f_n$ converges uniformly, but I don't have ideas how to do it.






real-analysis sequences-and-series analysis







share|cite|improve this question











$endgroup$





I have to prove that the sequence of functions
$$
f_n(x) = frac{n^2x}{1 + n^2x^2log n},
$$ converges.




So I think that I can just show that $lim_{n to infty} (f_n) = 0$ pointwise. But to show that the sequence does not converge uniformly, I need to show that $f_n$ has a point of maximum $x = frac{1}{n sqrt{log n}}$ and at this point, the limit is equal to infinity. Is that correct?



And finally I need to show that $forall alpha > 0$ in the interval $mathbb{R} setminus (-alpha,+alpha)$, the sequence $f_n$ converges uniformly, but I don't have ideas how to do it.






real-analysis sequences-and-series analysis




real-analysis sequences-and-series analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 21:33







user587192

















asked Dec 15 '18 at 21:24









Matheus FachiniMatheus Fachini

859




859












  • $begingroup$
    You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 21:39










  • $begingroup$
    Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
    $endgroup$
    – Matheus Fachini
    Dec 15 '18 at 21:45






  • 1




    $begingroup$
    Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 22:23


















  • $begingroup$
    You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 21:39










  • $begingroup$
    Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
    $endgroup$
    – Matheus Fachini
    Dec 15 '18 at 21:45






  • 1




    $begingroup$
    Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
    $endgroup$
    – Mindlack
    Dec 15 '18 at 22:23
















$begingroup$
You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
$endgroup$
– Mindlack
Dec 15 '18 at 21:39




$begingroup$
You do not need to prove that the limit is infinity, you just need to prove that the sequence does not go to zero.
$endgroup$
– Mindlack
Dec 15 '18 at 21:39












$begingroup$
Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
$endgroup$
– Matheus Fachini
Dec 15 '18 at 21:45




$begingroup$
Okay, I got it, for example $f_n(x) = x^n(1-x^n)$. When $x = sqrt[n]{1/2}$ we have $f_n = 1/4$, thus is not uniformly convergent, correct?
$endgroup$
– Matheus Fachini
Dec 15 '18 at 21:45




1




1




$begingroup$
Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
$endgroup$
– Mindlack
Dec 15 '18 at 22:23




$begingroup$
Yes, because the max-norm is always at least $1/4$ so cannot go to zero.
$endgroup$
– Mindlack
Dec 15 '18 at 22:23










1 Answer
1






active

oldest

votes


















2












$begingroup$

$1). $ If $x=0$ the sequence clearly converges to $0$. If not, then $|f_n(x)|le frac{1}{xlog n}to 0$ as $nto infty$ so $(f_n)$ is pointwise convergent.



$2). $ The trouble will be near zero, so take $x_n=1/n$. Then $x_nto 0$ whereas $f_n(x_n)to infty$, and so $(f_n)$ is not uniformly convergent.



$3). $ $f_n$ is odd for each $nge 1$ so it suffices to check $[alpha,infty).$ In this case,$ |f_n(x)|le frac{1}{alphalog n}$ so $(f_n)$ is uniformly bounded, and so converges uniformly.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Because $f_n(1/n)=n/(1+ln n)$
    $endgroup$
    – Matematleta
    Dec 16 '18 at 2:39










  • $begingroup$
    I had misread the numerator as $n^2x^2$.
    $endgroup$
    – Mark Viola
    Dec 16 '18 at 14:39











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3041979%2fprove-that-f-nx-fracn2x1-n2x2-log-n-converge-but-not-uniform%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

$1). $ If $x=0$ the sequence clearly converges to $0$. If not, then $|f_n(x)|le frac{1}{xlog n}to 0$ as $nto infty$ so $(f_n)$ is pointwise convergent.



$2). $ The trouble will be near zero, so take $x_n=1/n$. Then $x_nto 0$ whereas $f_n(x_n)to infty$, and so $(f_n)$ is not uniformly convergent.



$3). $ $f_n$ is odd for each $nge 1$ so it suffices to check $[alpha,infty).$ In this case,$ |f_n(x)|le frac{1}{alphalog n}$ so $(f_n)$ is uniformly bounded, and so converges uniformly.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Because $f_n(1/n)=n/(1+ln n)$
    $endgroup$
    – Matematleta
    Dec 16 '18 at 2:39










  • $begingroup$
    I had misread the numerator as $n^2x^2$.
    $endgroup$
    – Mark Viola
    Dec 16 '18 at 14:39
















2












$begingroup$

$1). $ If $x=0$ the sequence clearly converges to $0$. If not, then $|f_n(x)|le frac{1}{xlog n}to 0$ as $nto infty$ so $(f_n)$ is pointwise convergent.



$2). $ The trouble will be near zero, so take $x_n=1/n$. Then $x_nto 0$ whereas $f_n(x_n)to infty$, and so $(f_n)$ is not uniformly convergent.



$3). $ $f_n$ is odd for each $nge 1$ so it suffices to check $[alpha,infty).$ In this case,$ |f_n(x)|le frac{1}{alphalog n}$ so $(f_n)$ is uniformly bounded, and so converges uniformly.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Because $f_n(1/n)=n/(1+ln n)$
    $endgroup$
    – Matematleta
    Dec 16 '18 at 2:39










  • $begingroup$
    I had misread the numerator as $n^2x^2$.
    $endgroup$
    – Mark Viola
    Dec 16 '18 at 14:39














2












2








2





$begingroup$

$1). $ If $x=0$ the sequence clearly converges to $0$. If not, then $|f_n(x)|le frac{1}{xlog n}to 0$ as $nto infty$ so $(f_n)$ is pointwise convergent.



$2). $ The trouble will be near zero, so take $x_n=1/n$. Then $x_nto 0$ whereas $f_n(x_n)to infty$, and so $(f_n)$ is not uniformly convergent.



$3). $ $f_n$ is odd for each $nge 1$ so it suffices to check $[alpha,infty).$ In this case,$ |f_n(x)|le frac{1}{alphalog n}$ so $(f_n)$ is uniformly bounded, and so converges uniformly.






share|cite|improve this answer











$endgroup$



$1). $ If $x=0$ the sequence clearly converges to $0$. If not, then $|f_n(x)|le frac{1}{xlog n}to 0$ as $nto infty$ so $(f_n)$ is pointwise convergent.



$2). $ The trouble will be near zero, so take $x_n=1/n$. Then $x_nto 0$ whereas $f_n(x_n)to infty$, and so $(f_n)$ is not uniformly convergent.



$3). $ $f_n$ is odd for each $nge 1$ so it suffices to check $[alpha,infty).$ In this case,$ |f_n(x)|le frac{1}{alphalog n}$ so $(f_n)$ is uniformly bounded, and so converges uniformly.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 16 '18 at 4:51

























answered Dec 15 '18 at 22:24









MatematletaMatematleta

11.5k2920




11.5k2920












  • $begingroup$
    Because $f_n(1/n)=n/(1+ln n)$
    $endgroup$
    – Matematleta
    Dec 16 '18 at 2:39










  • $begingroup$
    I had misread the numerator as $n^2x^2$.
    $endgroup$
    – Mark Viola
    Dec 16 '18 at 14:39


















  • $begingroup$
    Because $f_n(1/n)=n/(1+ln n)$
    $endgroup$
    – Matematleta
    Dec 16 '18 at 2:39










  • $begingroup$
    I had misread the numerator as $n^2x^2$.
    $endgroup$
    – Mark Viola
    Dec 16 '18 at 14:39
















$begingroup$
Because $f_n(1/n)=n/(1+ln n)$
$endgroup$
– Matematleta
Dec 16 '18 at 2:39




$begingroup$
Because $f_n(1/n)=n/(1+ln n)$
$endgroup$
– Matematleta
Dec 16 '18 at 2:39












$begingroup$
I had misread the numerator as $n^2x^2$.
$endgroup$
– Mark Viola
Dec 16 '18 at 14:39




$begingroup$
I had misread the numerator as $n^2x^2$.
$endgroup$
– Mark Viola
Dec 16 '18 at 14:39


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3041979%2fprove-that-f-nx-fracn2x1-n2x2-log-n-converge-but-not-uniform%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Plaza Victoria

Image
Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
Read more

Puebla de Zaragoza

Image
Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc
Read more

Musa

Image
Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo
Read more