Area under the graph of $rmapstobinom nr$












2












$begingroup$


The question:




Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $rmapstobinom nr$, taking the $Gamma$-function definition of factorial.




Background: I'm just a high school student and I don't know enough calculus as to how to even approach a question like this. The question arose when my high school Maths teacher told us that there exists a function known as the $Gamma$-function which is an extension of the factorial function to all real numbers. I immediately had this question in mind since binomial theorem is all about factorials. I've taught myself some of the properties of the $Gamma$-function and learnt how to solve some definite integrals using the $Gamma$-function since then, but I just don't know how to approach this one. It has been nagging me ever since then. Even letting me know whether this is solvable or not (by hand and not computers) would be very helpful.






calculus integration definite-integrals area gamma-function







share|cite|improve this question











$endgroup$












  • $begingroup$
    This is a good question !
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 8:59










  • $begingroup$
    Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 9:09
















2












$begingroup$


The question:




Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $rmapstobinom nr$, taking the $Gamma$-function definition of factorial.




Background: I'm just a high school student and I don't know enough calculus as to how to even approach a question like this. The question arose when my high school Maths teacher told us that there exists a function known as the $Gamma$-function which is an extension of the factorial function to all real numbers. I immediately had this question in mind since binomial theorem is all about factorials. I've taught myself some of the properties of the $Gamma$-function and learnt how to solve some definite integrals using the $Gamma$-function since then, but I just don't know how to approach this one. It has been nagging me ever since then. Even letting me know whether this is solvable or not (by hand and not computers) would be very helpful.






calculus integration definite-integrals area gamma-function







share|cite|improve this question











$endgroup$












  • $begingroup$
    This is a good question !
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 8:59










  • $begingroup$
    Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 9:09














2












2








2





$begingroup$


The question:




Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $rmapstobinom nr$, taking the $Gamma$-function definition of factorial.




Background: I'm just a high school student and I don't know enough calculus as to how to even approach a question like this. The question arose when my high school Maths teacher told us that there exists a function known as the $Gamma$-function which is an extension of the factorial function to all real numbers. I immediately had this question in mind since binomial theorem is all about factorials. I've taught myself some of the properties of the $Gamma$-function and learnt how to solve some definite integrals using the $Gamma$-function since then, but I just don't know how to approach this one. It has been nagging me ever since then. Even letting me know whether this is solvable or not (by hand and not computers) would be very helpful.






calculus integration definite-integrals area gamma-function







share|cite|improve this question











$endgroup$




The question:




Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $rmapstobinom nr$, taking the $Gamma$-function definition of factorial.




Background: I'm just a high school student and I don't know enough calculus as to how to even approach a question like this. The question arose when my high school Maths teacher told us that there exists a function known as the $Gamma$-function which is an extension of the factorial function to all real numbers. I immediately had this question in mind since binomial theorem is all about factorials. I've taught myself some of the properties of the $Gamma$-function and learnt how to solve some definite integrals using the $Gamma$-function since then, but I just don't know how to approach this one. It has been nagging me ever since then. Even letting me know whether this is solvable or not (by hand and not computers) would be very helpful.






calculus integration definite-integrals area gamma-function




calculus integration definite-integrals area gamma-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 '18 at 9:10









Christoph

12k1642




12k1642










asked Dec 8 '18 at 8:21









user3611230user3611230

255




255












  • $begingroup$
    This is a good question !
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 8:59










  • $begingroup$
    Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 9:09


















  • $begingroup$
    This is a good question !
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 8:59










  • $begingroup$
    Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
    $endgroup$
    – Claude Leibovici
    Dec 8 '18 at 9:09
















$begingroup$
This is a good question !
$endgroup$
– Claude Leibovici
Dec 8 '18 at 8:59




$begingroup$
This is a good question !
$endgroup$
– Claude Leibovici
Dec 8 '18 at 8:59












$begingroup$
Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
$endgroup$
– Claude Leibovici
Dec 8 '18 at 9:09




$begingroup$
Working with high accuracy $I_5=31.3749$ and $I_{10}=1023.4546$. Quite close to $2 ^5=32$ and $2^{10}=1024$
$endgroup$
– Claude Leibovici
Dec 8 '18 at 9:09










1 Answer
1






active

oldest

votes


















5












$begingroup$

If I properly understand, you would like to compute
$$I_n=int_0^n binom{n}{r},dr=n!, int_0^n frac {dr} {(n-r)!,, r!}=Gamma (n+1), int_0^n frac{dr}{Gamma (r+1), Gamma (n-r+1)}$$ Unfortunately, there is no closed form even for
$$int_1^a {Gamma (r)},drqquad text{or} qquad int_1^a frac{dr}{Gamma (r)}$$ and you will be facing numerical integration.



For you curiosity, I give you below some values of $log_{10}(I_n)$ since $I_n$ varies extremely fast
$$left(
begin{array}{cc}
n & log_{10}(I_n) \
10 & 3.01007 \
20 & 6.02060 \
30 & 9.03090 \
40 & 12.0412 \
50 & 15.0515 \
60 & 18.0618 \
70 & 21.0721 \
80 & 24.0824 \
90 & 27.0927 \
100 & 30.1030
end{array}
right)$$ If you plot them, you could see that this is almost
$$log_{10}(I_n)= nlog_{10}(2)implies I_n sim 2^n$$ which is normal since
$$I_n sim sum_{r=0}^n binom{n}{r}=2^n$$






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%2f3030824%2farea-under-the-graph-of-r-mapsto-binom-nr%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









    5












    $begingroup$

    If I properly understand, you would like to compute
    $$I_n=int_0^n binom{n}{r},dr=n!, int_0^n frac {dr} {(n-r)!,, r!}=Gamma (n+1), int_0^n frac{dr}{Gamma (r+1), Gamma (n-r+1)}$$ Unfortunately, there is no closed form even for
    $$int_1^a {Gamma (r)},drqquad text{or} qquad int_1^a frac{dr}{Gamma (r)}$$ and you will be facing numerical integration.



    For you curiosity, I give you below some values of $log_{10}(I_n)$ since $I_n$ varies extremely fast
    $$left(
    begin{array}{cc}
    n & log_{10}(I_n) \
    10 & 3.01007 \
    20 & 6.02060 \
    30 & 9.03090 \
    40 & 12.0412 \
    50 & 15.0515 \
    60 & 18.0618 \
    70 & 21.0721 \
    80 & 24.0824 \
    90 & 27.0927 \
    100 & 30.1030
    end{array}
    right)$$     If you plot them, you could see that this is almost
    $$log_{10}(I_n)= nlog_{10}(2)implies I_n sim 2^n$$ which is normal since
    $$I_n sim sum_{r=0}^n binom{n}{r}=2^n$$






    share|cite|improve this answer









    $endgroup$


















      5












      $begingroup$

      If I properly understand, you would like to compute
      $$I_n=int_0^n binom{n}{r},dr=n!, int_0^n frac {dr} {(n-r)!,, r!}=Gamma (n+1), int_0^n frac{dr}{Gamma (r+1), Gamma (n-r+1)}$$ Unfortunately, there is no closed form even for
      $$int_1^a {Gamma (r)},drqquad text{or} qquad int_1^a frac{dr}{Gamma (r)}$$ and you will be facing numerical integration.



      For you curiosity, I give you below some values of $log_{10}(I_n)$ since $I_n$ varies extremely fast
      $$left(
      begin{array}{cc}
      n & log_{10}(I_n) \
      10 & 3.01007 \
      20 & 6.02060 \
      30 & 9.03090 \
      40 & 12.0412 \
      50 & 15.0515 \
      60 & 18.0618 \
      70 & 21.0721 \
      80 & 24.0824 \
      90 & 27.0927 \
      100 & 30.1030
      end{array}
      right)$$       If you plot them, you could see that this is almost
      $$log_{10}(I_n)= nlog_{10}(2)implies I_n sim 2^n$$ which is normal since
      $$I_n sim sum_{r=0}^n binom{n}{r}=2^n$$






      share|cite|improve this answer









      $endgroup$
















        5












        5








        5





        $begingroup$

        If I properly understand, you would like to compute
        $$I_n=int_0^n binom{n}{r},dr=n!, int_0^n frac {dr} {(n-r)!,, r!}=Gamma (n+1), int_0^n frac{dr}{Gamma (r+1), Gamma (n-r+1)}$$ Unfortunately, there is no closed form even for
        $$int_1^a {Gamma (r)},drqquad text{or} qquad int_1^a frac{dr}{Gamma (r)}$$ and you will be facing numerical integration.



        For you curiosity, I give you below some values of $log_{10}(I_n)$ since $I_n$ varies extremely fast
        $$left(
        begin{array}{cc}
        n & log_{10}(I_n) \
        10 & 3.01007 \
        20 & 6.02060 \
        30 & 9.03090 \
        40 & 12.0412 \
        50 & 15.0515 \
        60 & 18.0618 \
        70 & 21.0721 \
        80 & 24.0824 \
        90 & 27.0927 \
        100 & 30.1030
        end{array}
        right)$$         If you plot them, you could see that this is almost
        $$log_{10}(I_n)= nlog_{10}(2)implies I_n sim 2^n$$ which is normal since
        $$I_n sim sum_{r=0}^n binom{n}{r}=2^n$$






        share|cite|improve this answer









        $endgroup$



        If I properly understand, you would like to compute
        $$I_n=int_0^n binom{n}{r},dr=n!, int_0^n frac {dr} {(n-r)!,, r!}=Gamma (n+1), int_0^n frac{dr}{Gamma (r+1), Gamma (n-r+1)}$$ Unfortunately, there is no closed form even for
        $$int_1^a {Gamma (r)},drqquad text{or} qquad int_1^a frac{dr}{Gamma (r)}$$ and you will be facing numerical integration.



        For you curiosity, I give you below some values of $log_{10}(I_n)$ since $I_n$ varies extremely fast
        $$left(
        begin{array}{cc}
        n & log_{10}(I_n) \
        10 & 3.01007 \
        20 & 6.02060 \
        30 & 9.03090 \
        40 & 12.0412 \
        50 & 15.0515 \
        60 & 18.0618 \
        70 & 21.0721 \
        80 & 24.0824 \
        90 & 27.0927 \
        100 & 30.1030
        end{array}
        right)$$         If you plot them, you could see that this is almost
        $$log_{10}(I_n)= nlog_{10}(2)implies I_n sim 2^n$$ which is normal since
        $$I_n sim sum_{r=0}^n binom{n}{r}=2^n$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 8 '18 at 8:57









        Claude LeiboviciClaude Leibovici

        121k1157133




        121k1157133






























            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%2f3030824%2farea-under-the-graph-of-r-mapsto-binom-nr%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

            Le Mesnil-Réaume

            Image
            Le Mesnil-Réaume Region Normandie Département Seine-Maritime Arrondissement Dieppe Kanton Eu Gemeindeverband Villes Sœurs Koordinaten 49° 58′  N , 1° 27′  O 49.9725 1.4463888888889 Koordinaten: 49° 58′  N , 1° 27′  O Höhe 92–156 m Fläche 5,55 km 2 Einwohner 793 (1. Januar 2016) Bevölkerungsdichte 143 Einw./km 2 Postleitzahl 76260 INSEE-Code 76435 Website http://www.mesnil-reaume.net/ Rathaus (Mairie) und Schule von Le Mesnil-Réaume Le Mesnil-Réaume ist eine französische Gemeinde mit 793 Einwohnern (Stand: 1. Januar 2016) im Département Seine-Maritime in der Region Normandie (vor 2016 Haute-Normandie). Sie gehört zum Arrondissement Dieppe und zum Kanton Eu. Die Einwohner werden Mesnillais genannt. Inhaltsverzeichnis 1 Geographie 2 Bevölkerungsentwicklung 3 Sehenswürdigkeiten 4 Weblinks Geographie | Le Mesnil-Réaume liegt etwa 43 Kilomet
            Read more

            Ida-Boy-Ed-Garten

            Image
            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
            Read more

            web3.py web3.isConnected() returns false always

            Image
            1 I am trying an tutorial on web3.py so i created a profile in infura and used its project id.But it shows false always when i try web3.isConnected() PS C:UsersHP> python Python 3.6.3 (v3.6.3:2c5fed8, Oct 3 2017, 18:11:49) [MSC v.1900 64 bit (AMD64)] on win32 Type "help", "copyright", "credits" or "license" for more information. >>> from web3 import Web3 >>> infura_url = "mainnet.infura.io/v3/******************************" >>> web3 = Web3(Web3.HTTPProvider(infura_url)) >>> web3.isConnected() False web3.py share | improve this question
            Read more