Proving the uniform convergence of the average sequence of $f_n(x)=sin(nx)$












2












$begingroup$


I was asked to prove that:



1) $f_n(x)=sin(nx)$ does not converge pointwise.



2) The average sequence of $f_n(x)=sin(nx)$ is uniformly convergent.



I secceed to prove the first part but I cannot prove the other one. In addition it is not allowed to use the M-test.



Thanks. (I do not know how to use the function signs.)






sequences-and-series uniform-convergence







share|cite|improve this question











$endgroup$












  • $begingroup$
    I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 12:59










  • $begingroup$
    I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:19
















2












$begingroup$


I was asked to prove that:



1) $f_n(x)=sin(nx)$ does not converge pointwise.



2) The average sequence of $f_n(x)=sin(nx)$ is uniformly convergent.



I secceed to prove the first part but I cannot prove the other one. In addition it is not allowed to use the M-test.



Thanks. (I do not know how to use the function signs.)






sequences-and-series uniform-convergence







share|cite|improve this question











$endgroup$












  • $begingroup$
    I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 12:59










  • $begingroup$
    I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:19














2












2








2





$begingroup$


I was asked to prove that:



1) $f_n(x)=sin(nx)$ does not converge pointwise.



2) The average sequence of $f_n(x)=sin(nx)$ is uniformly convergent.



I secceed to prove the first part but I cannot prove the other one. In addition it is not allowed to use the M-test.



Thanks. (I do not know how to use the function signs.)






sequences-and-series uniform-convergence







share|cite|improve this question











$endgroup$




I was asked to prove that:



1) $f_n(x)=sin(nx)$ does not converge pointwise.



2) The average sequence of $f_n(x)=sin(nx)$ is uniformly convergent.



I secceed to prove the first part but I cannot prove the other one. In addition it is not allowed to use the M-test.



Thanks. (I do not know how to use the function signs.)






sequences-and-series uniform-convergence




sequences-and-series uniform-convergence






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 29 '18 at 12:50









Saad

20.1k92352




20.1k92352










asked Dec 29 '18 at 12:37









DANIEL SHALAMDANIEL SHALAM

115




115












  • $begingroup$
    I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 12:59










  • $begingroup$
    I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:19


















  • $begingroup$
    I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 12:59










  • $begingroup$
    I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:19
















$begingroup$
I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
$endgroup$
– SmileyCraft
Dec 29 '18 at 12:59




$begingroup$
I expect the only property of $sin(x)$ you will need is that it has an irrational period. The average will converge to the integral over the period.
$endgroup$
– SmileyCraft
Dec 29 '18 at 12:59












$begingroup$
I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:19




$begingroup$
I forgot to mention pi/3=>x>=2pi/3 . can you explain how it helps?
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:19










1 Answer
1






active

oldest

votes


















1












$begingroup$

Hint:



$$frac1nsum_{k=0}^{n-1} e^{ikx}=frac{e^{inx}-1}{n(e^{ix}-1)}.$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think you mean $sum_{i=0}^{n-1}e^{inx}$.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:00










  • $begingroup$
    We do not use complex numbers at our solutions..
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:21










  • $begingroup$
    You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:22










  • $begingroup$
    I think you do need an edge case for $x=2kpi$ by the way.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:29












  • $begingroup$
    I can not see how it helps yet... another hint can help.
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:32











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%2f3055804%2fproving-the-uniform-convergence-of-the-average-sequence-of-f-nx-sinnx%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









1












$begingroup$

Hint:



$$frac1nsum_{k=0}^{n-1} e^{ikx}=frac{e^{inx}-1}{n(e^{ix}-1)}.$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think you mean $sum_{i=0}^{n-1}e^{inx}$.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:00










  • $begingroup$
    We do not use complex numbers at our solutions..
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:21










  • $begingroup$
    You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:22










  • $begingroup$
    I think you do need an edge case for $x=2kpi$ by the way.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:29












  • $begingroup$
    I can not see how it helps yet... another hint can help.
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:32
















1












$begingroup$

Hint:



$$frac1nsum_{k=0}^{n-1} e^{ikx}=frac{e^{inx}-1}{n(e^{ix}-1)}.$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think you mean $sum_{i=0}^{n-1}e^{inx}$.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:00










  • $begingroup$
    We do not use complex numbers at our solutions..
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:21










  • $begingroup$
    You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:22










  • $begingroup$
    I think you do need an edge case for $x=2kpi$ by the way.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:29












  • $begingroup$
    I can not see how it helps yet... another hint can help.
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:32














1












1








1





$begingroup$

Hint:



$$frac1nsum_{k=0}^{n-1} e^{ikx}=frac{e^{inx}-1}{n(e^{ix}-1)}.$$






share|cite|improve this answer











$endgroup$



Hint:



$$frac1nsum_{k=0}^{n-1} e^{ikx}=frac{e^{inx}-1}{n(e^{ix}-1)}.$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 29 '18 at 13:45

























answered Dec 29 '18 at 12:57









Yves DaoustYves Daoust

131k676229




131k676229












  • $begingroup$
    I think you mean $sum_{i=0}^{n-1}e^{inx}$.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:00










  • $begingroup$
    We do not use complex numbers at our solutions..
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:21










  • $begingroup$
    You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:22










  • $begingroup$
    I think you do need an edge case for $x=2kpi$ by the way.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:29












  • $begingroup$
    I can not see how it helps yet... another hint can help.
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:32


















  • $begingroup$
    I think you mean $sum_{i=0}^{n-1}e^{inx}$.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:00










  • $begingroup$
    We do not use complex numbers at our solutions..
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:21










  • $begingroup$
    You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:22










  • $begingroup$
    I think you do need an edge case for $x=2kpi$ by the way.
    $endgroup$
    – SmileyCraft
    Dec 29 '18 at 13:29












  • $begingroup$
    I can not see how it helps yet... another hint can help.
    $endgroup$
    – DANIEL SHALAM
    Dec 29 '18 at 13:32
















$begingroup$
I think you mean $sum_{i=0}^{n-1}e^{inx}$.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:00




$begingroup$
I think you mean $sum_{i=0}^{n-1}e^{inx}$.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:00












$begingroup$
We do not use complex numbers at our solutions..
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:21




$begingroup$
We do not use complex numbers at our solutions..
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:21












$begingroup$
You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:22




$begingroup$
You can use Euler's formula to get back $sin$ and $cos$. The solution Yves is hinting at is a really neat use of complex numbers.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:22












$begingroup$
I think you do need an edge case for $x=2kpi$ by the way.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:29






$begingroup$
I think you do need an edge case for $x=2kpi$ by the way.
$endgroup$
– SmileyCraft
Dec 29 '18 at 13:29














$begingroup$
I can not see how it helps yet... another hint can help.
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:32




$begingroup$
I can not see how it helps yet... another hint can help.
$endgroup$
– DANIEL SHALAM
Dec 29 '18 at 13:32


















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%2f3055804%2fproving-the-uniform-convergence-of-the-average-sequence-of-f-nx-sinnx%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

Wiesbaden

Image
Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
Read more

Marschland

Image
Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
Read more

Dieringhausen

Image
Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
Read more