Difference between divergent series and series with no limit?












1












$begingroup$


Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09


















1












$begingroup$


Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09
















1












1








1





$begingroup$


Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series







share|cite|improve this question









$endgroup$




Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.






sequences-and-series




sequences-and-series






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jun 18 '13 at 19:02









Aaron BooherAaron Booher

814




814








  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09
















  • 1




    $begingroup$
    The limit could be $infty$ for example, in which case we still say it's divergent.
    $endgroup$
    – Najib Idrissi
    Jun 18 '13 at 19:04










  • $begingroup$
    If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
    $endgroup$
    – Ben Passer
    Jun 18 '13 at 21:09










1




1




$begingroup$
The limit could be $infty$ for example, in which case we still say it's divergent.
$endgroup$
– Najib Idrissi
Jun 18 '13 at 19:04




$begingroup$
The limit could be $infty$ for example, in which case we still say it's divergent.
$endgroup$
– Najib Idrissi
Jun 18 '13 at 19:04












$begingroup$
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
$endgroup$
– Ben Passer
Jun 18 '13 at 21:09






$begingroup$
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
$endgroup$
– Ben Passer
Jun 18 '13 at 21:09












2 Answers
2






active

oldest

votes


















0












$begingroup$

If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



The partial sums obtain a finite limit if and only if the series is convergent.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57



















0












$begingroup$

In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





  1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


  2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






share|cite|improve this answer











$endgroup$













    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%2f423863%2fdifference-between-divergent-series-and-series-with-no-limit%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57
















    0












    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57














    0












    0








    0





    $begingroup$

    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.






    share|cite|improve this answer









    $endgroup$



    If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.



    The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.



    The partial sums obtain a finite limit if and only if the series is convergent.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jun 18 '13 at 19:20









    Fly by NightFly by Night

    25.9k32978




    25.9k32978












    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57


















    • $begingroup$
      Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
      $endgroup$
      – Math1000
      May 27 '15 at 13:57
















    $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57




    $begingroup$
    Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
    $endgroup$
    – Math1000
    May 27 '15 at 13:57











    0












    $begingroup$

    In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





    1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


    2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





    Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





      1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


      2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





      Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





        1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


        2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





        Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.






        share|cite|improve this answer











        $endgroup$



        In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:





        1. A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).


        2. Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.





        Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 7 '17 at 15:04









        grg

        1,0431812




        1,0431812










        answered Apr 25 '15 at 7:56







        user223391





































            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%2f423863%2fdifference-between-divergent-series-and-series-with-no-limit%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

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more