If $sum_{n=1}^{infty} sum_{k=1}^{infty} f_{n}(k) = infty$ then $sum_{k=1}^{infty} sum_{n=1}^{infty} f_{n}(k)...












3












$begingroup$


Suppose ${f_{n}}_{n=1}^{infty}$ be functions such that $f_{n} : Bbb{N} rightarrow Bbb{R}^{+}$ for each $n$.



I was trying to prove -



If $sum_{n=1}^{infty} sum_{k=1}^{infty} f_{n}(k) = infty$ then $sum_{k=1}^{infty} sum_{n=1}^{infty} f_{n}(k) = infty$.



I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?



Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?






calculus functions convergence summation







share|cite|improve this question









$endgroup$












  • $begingroup$
    Values in $mathbb R^+$, yes. You should be able to prove it.
    $endgroup$
    – GEdgar
    Dec 31 '18 at 21:51










  • $begingroup$
    thank u for the hint!!
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:19
















3












$begingroup$


Suppose ${f_{n}}_{n=1}^{infty}$ be functions such that $f_{n} : Bbb{N} rightarrow Bbb{R}^{+}$ for each $n$.



I was trying to prove -



If $sum_{n=1}^{infty} sum_{k=1}^{infty} f_{n}(k) = infty$ then $sum_{k=1}^{infty} sum_{n=1}^{infty} f_{n}(k) = infty$.



I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?



Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?






calculus functions convergence summation







share|cite|improve this question









$endgroup$












  • $begingroup$
    Values in $mathbb R^+$, yes. You should be able to prove it.
    $endgroup$
    – GEdgar
    Dec 31 '18 at 21:51










  • $begingroup$
    thank u for the hint!!
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:19














3












3








3





$begingroup$


Suppose ${f_{n}}_{n=1}^{infty}$ be functions such that $f_{n} : Bbb{N} rightarrow Bbb{R}^{+}$ for each $n$.



I was trying to prove -



If $sum_{n=1}^{infty} sum_{k=1}^{infty} f_{n}(k) = infty$ then $sum_{k=1}^{infty} sum_{n=1}^{infty} f_{n}(k) = infty$.



I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?



Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?






calculus functions convergence summation







share|cite|improve this question









$endgroup$




Suppose ${f_{n}}_{n=1}^{infty}$ be functions such that $f_{n} : Bbb{N} rightarrow Bbb{R}^{+}$ for each $n$.



I was trying to prove -



If $sum_{n=1}^{infty} sum_{k=1}^{infty} f_{n}(k) = infty$ then $sum_{k=1}^{infty} sum_{n=1}^{infty} f_{n}(k) = infty$.



I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?



Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?






calculus functions convergence summation




calculus functions convergence summation






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 31 '18 at 21:37









BAYMAXBAYMAX

2,93021225




2,93021225












  • $begingroup$
    Values in $mathbb R^+$, yes. You should be able to prove it.
    $endgroup$
    – GEdgar
    Dec 31 '18 at 21:51










  • $begingroup$
    thank u for the hint!!
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:19


















  • $begingroup$
    Values in $mathbb R^+$, yes. You should be able to prove it.
    $endgroup$
    – GEdgar
    Dec 31 '18 at 21:51










  • $begingroup$
    thank u for the hint!!
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:19
















$begingroup$
Values in $mathbb R^+$, yes. You should be able to prove it.
$endgroup$
– GEdgar
Dec 31 '18 at 21:51




$begingroup$
Values in $mathbb R^+$, yes. You should be able to prove it.
$endgroup$
– GEdgar
Dec 31 '18 at 21:51












$begingroup$
thank u for the hint!!
$endgroup$
– BAYMAX
Dec 31 '18 at 22:19




$begingroup$
thank u for the hint!!
$endgroup$
– BAYMAX
Dec 31 '18 at 22:19










1 Answer
1






active

oldest

votes


















2












$begingroup$

With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $infty$,



$$sum_{n=1}^N sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^N f_{nk}$$ and by the MCT



$$sum_{n=1}^infty sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^infty f_{nk}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:08






  • 1




    $begingroup$
    @BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:12






  • 1




    $begingroup$
    $sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:13






  • 1




    $begingroup$
    Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:24






  • 1




    $begingroup$
    The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:26











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%2f3058059%2fif-sum-n-1-infty-sum-k-1-infty-f-nk-infty-then-sum-k-1%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












$begingroup$

With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $infty$,



$$sum_{n=1}^N sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^N f_{nk}$$ and by the MCT



$$sum_{n=1}^infty sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^infty f_{nk}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:08






  • 1




    $begingroup$
    @BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:12






  • 1




    $begingroup$
    $sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:13






  • 1




    $begingroup$
    Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:24






  • 1




    $begingroup$
    The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:26
















2












$begingroup$

With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $infty$,



$$sum_{n=1}^N sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^N f_{nk}$$ and by the MCT



$$sum_{n=1}^infty sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^infty f_{nk}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:08






  • 1




    $begingroup$
    @BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:12






  • 1




    $begingroup$
    $sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:13






  • 1




    $begingroup$
    Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:24






  • 1




    $begingroup$
    The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:26














2












2








2





$begingroup$

With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $infty$,



$$sum_{n=1}^N sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^N f_{nk}$$ and by the MCT



$$sum_{n=1}^infty sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^infty f_{nk}$$






share|cite|improve this answer









$endgroup$



With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $infty$,



$$sum_{n=1}^N sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^N f_{nk}$$ and by the MCT



$$sum_{n=1}^infty sum_{k=1}^infty f_{nk} = sum_{k=1}^infty sum_{n=1}^infty f_{nk}$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 31 '18 at 21:54









RRLRRL

53k42573




53k42573












  • $begingroup$
    By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:08






  • 1




    $begingroup$
    @BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:12






  • 1




    $begingroup$
    $sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:13






  • 1




    $begingroup$
    Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:24






  • 1




    $begingroup$
    The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:26


















  • $begingroup$
    By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
    $endgroup$
    – BAYMAX
    Dec 31 '18 at 22:08






  • 1




    $begingroup$
    @BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:12






  • 1




    $begingroup$
    $sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:13






  • 1




    $begingroup$
    Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:24






  • 1




    $begingroup$
    The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
    $endgroup$
    – RRL
    Dec 31 '18 at 22:26
















$begingroup$
By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
$endgroup$
– BAYMAX
Dec 31 '18 at 22:08




$begingroup$
By applying Monotone Convergence Theorem, it involves integrals and limit exchange right? here there are two summations?
$endgroup$
– BAYMAX
Dec 31 '18 at 22:08




1




1




$begingroup$
@BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
$endgroup$
– RRL
Dec 31 '18 at 22:12




$begingroup$
@BAYMAX: There is a very general way by connecting sums to integrals with counting measure but it is not necessary. There is a monotone convergence theorem just for switching limits with sums.
$endgroup$
– RRL
Dec 31 '18 at 22:12




1




1




$begingroup$
$sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
$endgroup$
– RRL
Dec 31 '18 at 22:13




$begingroup$
$sum_{n=1}^Nf_{nk}$ is an increasing sequence with respect to $N$ because each term you add is nonnegative.
$endgroup$
– RRL
Dec 31 '18 at 22:13




1




1




$begingroup$
Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
$endgroup$
– RRL
Dec 31 '18 at 22:24




$begingroup$
Here is the proof. Notice that $lim_{N to infty} sum_{k=1}^infty sum_{n=1}^N f_{nk} = lim_{N to infty} sum_{k=1}^infty S_{Nk} = sum_{k = 1}^infty lim_{N to infty} S_{Nk}$ since the partial sums $S_{Nk}$ are monotone increasing for each $k$.
$endgroup$
– RRL
Dec 31 '18 at 22:24




1




1




$begingroup$
The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
$endgroup$
– RRL
Dec 31 '18 at 22:26




$begingroup$
The key thing here is the first result -- finite sum of limits equals limit of finite sum -- when everything is nonnegative even if one of the limits is $+infty$.
$endgroup$
– RRL
Dec 31 '18 at 22:26


















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%2f3058059%2fif-sum-n-1-infty-sum-k-1-infty-f-nk-infty-then-sum-k-1%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