Convergence of $ sum_{n=0}^{infty}{ left( -1 right)^n} $












-3












$begingroup$


Let’s consider well known problem of sum of alternate series of positive and negative ones.
If we look at it from “back end” we may write 0 as an infinite sum where we use fact that $1-1$ is 0:



$0=0+0+ldots=left(1-1right)+left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



We may do same calculation starting with one



$1=1-0-0-ldots=1-left(1-1right)-left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



Additionally, we may derive following third results:



$S=sum_{n=0}^{infty}left(-1right)^n$



$S=1-1+1-1+1-1+ldots$



$2S=1-1+1-1+1-1+ldots $



$space spacespacespacespacespacespacespacespacespacespacespace +1-1+1-1+1-ldots $



$2S=1 $



$S=frac{1}{2}$



This leads to contradiction:



$left(S=0 land S=1land S=frac{1}{2}right) Rightarrowleft(0=1=frac{1}{2}right)$



Additionally, the third result is out of the integer numbers ring, however we used only summation in the sum. I suppose that the problem occurs when we omit the parenthesis in terms (1-1). We make a sum of two infinite series 1+1+1+… and -1-1-1-... which do not converge.
However, it does not seem that we violate any arithmetic rule of integer numbers in the intermediate steps.



I wonder, if it means that Peano axioms which define the natural numbers (and then definition of integers is based on the natural numbers), are not enough to work with infinite series. What axioms do we need else (beyond induction)? How does it relate to Goedel theorems?






sequences-and-series







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    The problem is that you work with an undefined object. That's all.
    $endgroup$
    – amsmath
    Jan 26 at 21:42










  • $begingroup$
    Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
    $endgroup$
    – rtybase
    Jan 26 at 22:00
















-3












$begingroup$


Let’s consider well known problem of sum of alternate series of positive and negative ones.
If we look at it from “back end” we may write 0 as an infinite sum where we use fact that $1-1$ is 0:



$0=0+0+ldots=left(1-1right)+left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



We may do same calculation starting with one



$1=1-0-0-ldots=1-left(1-1right)-left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



Additionally, we may derive following third results:



$S=sum_{n=0}^{infty}left(-1right)^n$



$S=1-1+1-1+1-1+ldots$



$2S=1-1+1-1+1-1+ldots $



$space spacespacespacespacespacespacespacespacespacespacespace +1-1+1-1+1-ldots $



$2S=1 $



$S=frac{1}{2}$



This leads to contradiction:



$left(S=0 land S=1land S=frac{1}{2}right) Rightarrowleft(0=1=frac{1}{2}right)$



Additionally, the third result is out of the integer numbers ring, however we used only summation in the sum. I suppose that the problem occurs when we omit the parenthesis in terms (1-1). We make a sum of two infinite series 1+1+1+… and -1-1-1-... which do not converge.
However, it does not seem that we violate any arithmetic rule of integer numbers in the intermediate steps.



I wonder, if it means that Peano axioms which define the natural numbers (and then definition of integers is based on the natural numbers), are not enough to work with infinite series. What axioms do we need else (beyond induction)? How does it relate to Goedel theorems?






sequences-and-series







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    The problem is that you work with an undefined object. That's all.
    $endgroup$
    – amsmath
    Jan 26 at 21:42










  • $begingroup$
    Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
    $endgroup$
    – rtybase
    Jan 26 at 22:00














-3












-3








-3





$begingroup$


Let’s consider well known problem of sum of alternate series of positive and negative ones.
If we look at it from “back end” we may write 0 as an infinite sum where we use fact that $1-1$ is 0:



$0=0+0+ldots=left(1-1right)+left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



We may do same calculation starting with one



$1=1-0-0-ldots=1-left(1-1right)-left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



Additionally, we may derive following third results:



$S=sum_{n=0}^{infty}left(-1right)^n$



$S=1-1+1-1+1-1+ldots$



$2S=1-1+1-1+1-1+ldots $



$space spacespacespacespacespacespacespacespacespacespacespace +1-1+1-1+1-ldots $



$2S=1 $



$S=frac{1}{2}$



This leads to contradiction:



$left(S=0 land S=1land S=frac{1}{2}right) Rightarrowleft(0=1=frac{1}{2}right)$



Additionally, the third result is out of the integer numbers ring, however we used only summation in the sum. I suppose that the problem occurs when we omit the parenthesis in terms (1-1). We make a sum of two infinite series 1+1+1+… and -1-1-1-... which do not converge.
However, it does not seem that we violate any arithmetic rule of integer numbers in the intermediate steps.



I wonder, if it means that Peano axioms which define the natural numbers (and then definition of integers is based on the natural numbers), are not enough to work with infinite series. What axioms do we need else (beyond induction)? How does it relate to Goedel theorems?






sequences-and-series







share|cite|improve this question











$endgroup$




Let’s consider well known problem of sum of alternate series of positive and negative ones.
If we look at it from “back end” we may write 0 as an infinite sum where we use fact that $1-1$ is 0:



$0=0+0+ldots=left(1-1right)+left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



We may do same calculation starting with one



$1=1-0-0-ldots=1-left(1-1right)-left(1-1right) +ldots=1-1+1-1+1-1+ldots=sum_{n=0}^{infty}{left(-1right)^n=S}$



Additionally, we may derive following third results:



$S=sum_{n=0}^{infty}left(-1right)^n$



$S=1-1+1-1+1-1+ldots$



$2S=1-1+1-1+1-1+ldots $



$space spacespacespacespacespacespacespacespacespacespacespace +1-1+1-1+1-ldots $



$2S=1 $



$S=frac{1}{2}$



This leads to contradiction:



$left(S=0 land S=1land S=frac{1}{2}right) Rightarrowleft(0=1=frac{1}{2}right)$



Additionally, the third result is out of the integer numbers ring, however we used only summation in the sum. I suppose that the problem occurs when we omit the parenthesis in terms (1-1). We make a sum of two infinite series 1+1+1+… and -1-1-1-... which do not converge.
However, it does not seem that we violate any arithmetic rule of integer numbers in the intermediate steps.



I wonder, if it means that Peano axioms which define the natural numbers (and then definition of integers is based on the natural numbers), are not enough to work with infinite series. What axioms do we need else (beyond induction)? How does it relate to Goedel theorems?






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








edited Jan 26 at 21:39









Rhys Hughes

7,0351630




7,0351630










asked Jan 26 at 21:37









ArekArek

174




174








  • 5




    $begingroup$
    The problem is that you work with an undefined object. That's all.
    $endgroup$
    – amsmath
    Jan 26 at 21:42










  • $begingroup$
    Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
    $endgroup$
    – rtybase
    Jan 26 at 22:00














  • 5




    $begingroup$
    The problem is that you work with an undefined object. That's all.
    $endgroup$
    – amsmath
    Jan 26 at 21:42










  • $begingroup$
    Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
    $endgroup$
    – rtybase
    Jan 26 at 22:00








5




5




$begingroup$
The problem is that you work with an undefined object. That's all.
$endgroup$
– amsmath
Jan 26 at 21:42




$begingroup$
The problem is that you work with an undefined object. That's all.
$endgroup$
– amsmath
Jan 26 at 21:42












$begingroup$
Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
$endgroup$
– rtybase
Jan 26 at 22:00




$begingroup$
Good example of how far one can go, simply by ignoring a simple definition (of convergence in this case) :)
$endgroup$
– rtybase
Jan 26 at 22:00










1 Answer
1






active

oldest

votes


















1












$begingroup$

You are dealing with a divergent series as if it was convergnt. So, no wonder that you reach a contradiction. And, no, this has nothing to do with Gödel's theorem.






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%2f3088797%2fconvergence-of-sum-n-0-infty-left-1-rightn%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












    $begingroup$

    You are dealing with a divergent series as if it was convergnt. So, no wonder that you reach a contradiction. And, no, this has nothing to do with Gödel's theorem.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      You are dealing with a divergent series as if it was convergnt. So, no wonder that you reach a contradiction. And, no, this has nothing to do with Gödel's theorem.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        You are dealing with a divergent series as if it was convergnt. So, no wonder that you reach a contradiction. And, no, this has nothing to do with Gödel's theorem.






        share|cite|improve this answer









        $endgroup$



        You are dealing with a divergent series as if it was convergnt. So, no wonder that you reach a contradiction. And, no, this has nothing to do with Gödel's theorem.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 26 at 21:53









        José Carlos SantosJosé Carlos Santos

        169k23132237




        169k23132237






























            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%2f3088797%2fconvergence-of-sum-n-0-infty-left-1-rightn%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

            MongoDB - Not Authorized To Execute Command

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
            Read more

            in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

            Image
            1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
            Read more

            How to fix TextFormField cause rebuild widget in Flutter

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I've got the same issue at #9280. I tried all solutions from that topic with/without GlobalKey for Scaffold and Form but the problem wasn't fixed. I tried to build a authentication form with login and signup. But the keyboard doesn't show when I tap the TextFormField at the first time. Then I try to input some letter from physical keyboard, the widget is rebuilt. Like this: login screen Nothing to be log when I run flutter run --verbose, so I logged manually. Every tapping on field, the widget called dispose -> init -> build. [√] Flutter (Channel beta, v1.0.0, on Microsoft Windows [Version 10.0.17763.195], ...
            Read more