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







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$endgroup$












  • $begingroup$
    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






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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









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    1 Answer
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    1 Answer
    1






    active

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    active

    oldest

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    active

    oldest

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    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






























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