A beautiful curve that we can find in the expansion of  (((a + b))^n)

Yasushi Kotera
Ryohei Miyadera      
Kwansei Gakuin high school

1. Introduction

When we used Mathematica to expand (((a + b))^n), some high school students found the beautiful curve.  We tried to find the limit of the curve when we make n⟶∞, and found out that the limit of the curve was the graph of an elementary function.  Our main result is Theorem 1 in this paper.

Example 1.The expansion of(((a + b))^20) . If we expand(((a + b))^20),then we will find a beautiful curve.

Expand[(((a + b))^100)]

Example 2.  The expansion of (((a + b))^20).
If we expand (((a + b))^20), then  we will find a beautiful curve.

Clear[a, b];
g[n_] := 
      TableForm[Table[Binomial[n, k] (a^(n - k)) (b^k), {k, 0, n}]]; 
    g[20]

Example 3.  The expansion of (((a + b))^n) for n = 2, 4, 8, 16, 32, 64.
We use the binomial theorem to study the expansion of (((a + b))^n).  It is easier to appreciate the beauty of the expansion by this way. The shapes seem to be getting nearer to a certain shape.

Clear[a, b];
g[n_] := 
        TableForm[
          Table[Binomial[n, k] (a^(n - k)) (b^k), {k, 0, 
              n}]];
k = {2, 4, 8, 16, 32, 64};
\(Do[
Print["If we expand" \((a + b)\)\^k[\([n]\)]]; 
Print[", then we get"]; 
Print[g[k[\([n]\)]]
];, {n, 1, 
        6}]\)

Example 4.  The shapes of (((a + b))^n) divided by  n for n = 3, 6, 9, 12, ⋯,75.  It is easy to see that these shapes are reaching a certain limit.
Of course it is difficult to compare curves with  much  difference in size, so we compare them by making their size almost the same. We can do this by dividing the above expansion (((a + b))^n) by the number  n.  The expansion of (((a + b))^n)  can be represented by the list  {(Subscript[, n])(Subscript[C, 0]), (Subscript[, n])(Subscript[C, 1]),⋯,(Subscript[, n])(Subscript[C, m])}, and if we divide the shape of the expansion by  number  n  , then it is given by the list  { {(0/n),(Subscript[, n])(((Subscript[C, 0]))/n)},{(1/n),(Subscript[, n])(((Subscript[C, 1]))/n)},{(2/n),(Subscript[, n])(((Subscript[C, 2]))/n)}, ⋯,{(m/n),(Subscript[, n])(((Subscript[C, m]))/n)}, ⋯,{(n/n),(Subscript[, n])(((Subscript[C, n]))/n)}}　using the x-y coordinates.

f[n_Integer] := 
      ListPlot[Table[{k, Length[IntegerDigits[Binomial[3  n, k]]]}, {k, 0, 
                3  n}]/(3  n) // N, PlotRange -> {0, 1}, 
        PlotStyle -> RGBColor[0, 0, 1]]; 
    Do[Print[n]; f[n], {n, 1, 25}]

Lemma 1.  If we make  n and m big enough while keeping  (m/n) as an almost the same number, then we have    \(\(\(Log\_10\) \(\_n\)\(C\_m\)\)\/n\)∼-((m/n) (Subscript[log, 10]) 
                  (m/n) + (1 - (m/n)) (Subscript[log, 10]) (1 - 
                      (m/n))).

Proof.  In this proof we assume that   n  and  m  are big enough and we can use  Sterling's formula  for n and  m. \(\(\_n\)\(C\_m\) = \(n!\)\/\(\(\((n - m)\)!\) \(m!\)\)\)∼(((Sqrt[2  π]) (n^(n + (1/2))) 
              (e^(-n)))/((Sqrt[2  π]) (((n - m))^(n - m + 
                    (1/2))) (e^(-(n - 
                        m))) (Sqrt[2  π]) (m^(m + (1/2))) 
              (e^(-m)))) =((n^(n + (1/2)))/((Sqrt[2  π]) (((n - 
                      m))^(n - m + (1/2))) (m^(m + (1/2)))))=(((n^(n - m + (1/2))) 
              (n^m))/((Sqrt[2  π]) (((n - m))^(n - m + (1/2))) 
              (m^(m + (1/2)))))\(\(\(=\)\(\(1\/\@\(2  π\)\) \(\((n\/\(n - m\))\)\^\(n - m + 
                1\/2\)\) \(\((n\/m)\)\^m\) \(\(1\/m\^\(1\/2\)\)\(.\)\)\)\)\)
\(\(\(If\ \ m\/n\ is\ an\ almost\ the\ same\ number\)\(,\)\(\ \)\(then\)\(\ \)\)\)(((Subscript[Log, 10]) (Subscript[, n])(Subscript[C, m]))/n)\(\(\(∼\)\(\((\(n - m\)\/n + 
                1\/\(2  n\))\) \(Log\_10\) \((n\/\(n - m\))\) + \(m\/n\) \(Log\_10\) \((n\/m)\) - \(\(Log\_10\) \@\(2  π\)\)\/n - \(1\/\(2  
                  n\)\) \(Log\_10\) m\)\)\)\(\(\(∼\)\(\(\(n - 
                  m\)\/n\) \(Log\_10\) \((n\/\(n - m\))\) + \(m\/n\) \(Log\_10\) \((n\/m)\)\)\)\)
\(\(\(∼\)\(-\((\(\(n - 
                      m\)\/n\) \(Log\_10\) \((\(n - m\)\/n)\) + \(m\/n\) \(Log\_10\) \((m\/n)\))\)\)\)\).

Theorem 1.  The shapes of the expansion of  (((a + b))^n)  will get near to the graph of the funcion f (x) = -((Subscript[xlog, 10]) 
                x + (1 - x) (Subscript[log, 10]) (1 - x)),as n->∞.

As in the Example 3  we divide the shape of the expansion by  number  n  , and  it is given by the list  { {(0/n),(Subscript[, n])(((Subscript[C, 0]))/n)},{(1/n),(Subscript[, n])(((Subscript[C, 1]))/n)},{(2/n),(Subscript[, n])(((Subscript[C, 2]))/n)}, ⋯,{(m/n),(Subscript[, n])(((Subscript[C, m]))/n)}, ⋯,{(n/n),(Subscript[, n])(((Subscript[C, n]))/n)}}　using the x-y coordinates.  Therefore by Lemma 1 the shape of the expansion divided by the number  n  gets nearer and nearer to the graph of f (x) = -((Subscript[xlog, 10]) 
                x + (1 - x) (Subscript[log, 10]) (1 - x)),as n->∞.

Example 4.  The following is the graph of the function  f (x) = -((Subscript[xlog, 10]) 
                  x + (1 - x) (Subscript[log, 10]) (1 - x)).  Please compare this to the graph of the lists in Example 3.

Plot[-(x*Log[10, x] + (1 - x)*Log[10, 1 - x]), {x, 0, 1}, 
      PlotRange -> {0, 1}]

References.

R .  Miyadera and Y .  Kotera [2] Una Bella Curva Che Troviamo in Connessione con lo Sviluppo di  (((a + b))^n) Archimede 2/2005.