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

To store a contact into the json file from server.js file using a class in NodeJS

Image
0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
Read more

Redirect URL with Chrome Remote Debugging Android Devices

Image
1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
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