加算の順を工夫するの観察
ver 0.001 2022-10-3

更新履歴
ver 0.001 2022-10- 3 新規作成
 

まず、ガウスが、１〜ｎの和を加算の順序を工夫することによって、うまく処 理したのでそれを考えてみましょう。

第1節　前後3項の和

  1         1       1

------- + ----- + -----

 (n-1)^2   n^2    (n+1)^2



を考えてみましょ う。

(%i2) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);

                                      4
                                   3 n  + 1
(%o2)                        --------------------
                                    2  2        2
                             (n - 1)  n  (n + 1)
となりました。

nはn=3k-1なので、代入すると、

(%i38) n:3*k-1;
(%o38)                              3 k - 1
(%i39) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                           4        3        2
                      243 k  - 324 k  + 162 k  - 36 k + 4
(%o39)                -----------------------------------
                             2          2          2
                          9 k  (3 k - 2)  (3 k - 1)

分母に着目すると、(3k-2)^2(3k-1)^2において、kが偶数な ら、k=2jなので、

(%i51) k:2*j;
(%o51)                                2 j
(%i52) factor((3*k-2)^2*(3*k-1)^2);
                                       2          2
(%o52)                      4 (3 j - 1)  (6 j - 1)

kが奇数なら、k=2j+1なので、

(%i53) k:2*j+1;
(%o53)                              2 j + 1
(%i54) factor((3*k-2)^2*(3*k-1)^2);
                                       2          2
(%o54)                      4 (3 j + 1)  (6 j + 1)
よって、分母は、2^2と3^2を共通に持つ。

したがって、kが偶数なら、

(%i65) k:2*j;
(%o65)                                2 j
(%i66) n:3*k-1;
(%o66)                              6 j - 1
(%i67) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                           4        3        2
                      972 j  - 648 j  + 162 j  - 18 j + 1
(%o67)                -----------------------------------
                              2          2          2
                          36 j  (3 j - 1)  (6 j - 1)

kが奇数なら、

(%i68) k:2*j+1;
(%o68)                              2 j + 1
(%i69) n:3*k-1;
(%o69)                          3 (2 j + 1) - 1
(%i70) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                         4         3         2
                   3888 j  + 5184 j  + 2592 j  + 576 j + 49
(%o70)             ----------------------------------------
                                 2          2          2
                     36 (2 j + 1)  (3 j + 1)  (6 j + 1)
では、計算してみましょう。

(%i72) j:0;
(%o72)                                 0
(%i73) k:2*j+1;
(%o73)                                 1
(%i74) n:3*k-1;
(%o74)                                 2
(%i75) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                       2
                                      7
(%o75)                               -----
                                      2  2
                                     2  3
(%i76) j:1;
(%o76)                                 1
(%i77) k:2*j;
(%o77)                                 2
(%i78) n:3*k-1;
(%o78)                                 5
(%i79) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                     7 67
(%o79)                             --------
                                    4  2  2
                                   2  3  5
(%i80) k:2*j+1;
(%o80)                                 3
(%i81) n:3*k-1;
(%o81)                                 8
(%i82) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                    12289
(%o82)                             --------
                                    6  4  2
                                   2  3  7
(%i83) j:2;
(%o83)                                 2
(%i84) k:2*j;
(%o84)                                 4
(%i85) n:3*k-1;
(%o85)                                11
(%i86) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                    79 139
(%o86)                           ------------
                                  4  2  2   2
                                 2  3  5  11
(%i87) k:2*j+1;
(%o87)                                 5
(%i88) n:3*k-1;
(%o88)                                14
(%i89) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                    115249
(%o89)                          ---------------
                                 2  2  2  2   2
                                2  3  5  7  13
(%i90) j:3;
(%o90)                                 3
(%i91) k:2*j;
(%o91)                                 6
(%i92) n:3*k-1;
(%o92)                                17
(%i93) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                    37 1693
(%o93)                             ---------
                                    8  4   2
                                   2  3  17
(%i94) k:2*j+1;
(%o94)                                 7
(%i95) n:3*k-1;
(%o95)                                20
(%i96) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                   37 12973
(%o96)                          ---------------
                                 4  2  2  2   2
                                2  3  5  7  19
(%i97)
合計してみると、
(%i97) float(%o75+%o79+%o82+%o86+%o89+%o93+%o96);
(%o97)                         1.598430817609168

さらに、計算を続けると、

(%i98) j:4;
(%o98)                                 4
(%i99) k:2*j;
(%o99)                                 8
(%i100) n:3*k-1;
(%o100)                               23
(%i101) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                    7 29983
(%o101)                          -------------
                                  6  2   2   2
                                 2  3  11  23
(%i102) k:2*j+1;
(%o102)                                9
(%i103) n:3*k-1;
(%o103)                               26
(%i104) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                  7 151 1297
(%o104)                          ------------
                                  2  6  4   2
                                 2  3  5  13
(%i105) j:5;
(%o105)                                5
(%i106) k:2*j;
(%o106)                               10
(%i107) n:3*k-1;
(%o107)                               29
(%i108) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                   19 27919
(%o108)                         ---------------
                                 4  2  2  2   2
                                2  3  5  7  29
(%i109) k:2*j+1;
(%o109)                               11
(%i110) n:3*k-1;
(%o110)                               32
(%i111) factor(1/(n-1)^2+1/n^2+1/(n+1)^2);
                                   727 4327
(%o111)                         --------------
                                 10  2   2   2
                                2   3  11  31
(%i112)

昔の数学者は、素因数分解は、容易でなかった。すると、素因数分解で、攻め るという手があるかもしれない。

第2節　前後5項の和

  1       1         1       1        1

-------+------- + ----- + -------+-------

(n-2)^2 (n-1)^2   n^2    (n+1)^2  (n+2)^2



を考えてみましょ う。

(%i1) factor(1/(n-2)^2+1/(n-1)^2+1/n^2+1/(n+1)^2+1/(n+2)^2);
                             8       6       4
                          5 n  - 20 n  + 35 n  + 16
(%o1)               --------------------------------------
                           2        2  2        2        2
                    (n - 2)  (n - 1)  n  (n + 1)  (n + 2)


第 ３節　前後７項の和

  1       1       1         1       1        1       1

-------+-------+------- + ----- + -------+-------+-------

(n-3)^2 (n-2)^2 (n-1)^2   n^2    (n+1)^2  (n+2)^2  (n+3)^2



を考えてみましょ う。

(%i2) factor(1/(n-3)^2+1/(n-2)^2+1/(n-1)^2+1/n^2+1/(n+1)^2+1/(n+2)^2+1/(n+3)^2);
                12        10        8         6         4
             7 n   - 112 n   + 686 n  - 1736 n  + 2163 n  + 1296
(%o2)      --------------------------------------------------------
                  2        2        2  2        2        2        2
           (n - 3)  (n - 2)  (n - 1)  n  (n + 1)  (n + 2)  (n + 3)


第４節　中心の近い対称項の和

さて、それでは、n=1から、2n+1の中心から折返しを考えてみましょ う。中心を n とすると、

(%i5) factor(1/(n-1)^2+1/(n+1)^2);
                                      2
                                  2 (n  + 1)
(%o5)                          -----------------
                                      2        2
                               (n - 1)  (n + 1)
(%i4) factor(1/(n-2)^2+1/(n+2)^2);
                                      2
                                  2 (n  + 4)
(%o4)                          -----------------
                                      2        2
                               (n - 2)  (n + 2)
(%i3) factor(1/(n-3)^2+1/(n+3)^2);
                                      2
                                  2 (n  + 9)
(%o3)                          -----------------
                                      2        2
                               (n - 3)  (n + 3)
(%i6) factor(1/(n-4)^2+1/(n+4)^2);
                                      2
                                  2 (n  + 16)
(%o6)                          -----------------
                                      2        2
                               (n - 4)  (n + 4)
(%i7) factor(1/(n-5)^2+1/(n+5)^2);
                                      2
                                  2 (n  + 25)
(%o7)                          -----------------
                                      2        2
                               (n - 5)  (n + 5)
(%i8)
(%i8) factor(1/(n-6)^2+1/(n+6)^2);
                                      2
                                  2 (n  + 36)
(%o8)                          -----------------
                                      2        2
                               (n - 6)  (n + 6)
(%i9) factor(1/(n-7)^2+1/(n+7)^2);
                                      2
                                  2 (n  + 49)
(%o9)                          -----------------
                                      2        2
                               (n - 7)  (n + 7)
(%i10) factor(1/(n-8)^2+1/(n+8)^2);
                                      2
                                  2 (n  + 64)
(%o10)                         -----------------
                                      2        2
                               (n - 8)  (n + 8)
(%i11) factor(1/(n-9)^2+1/(n+9)^2);
                                      2
                                  2 (n  + 81)
(%o11)                         -----------------
                                      2        2
                               (n - 9)  (n + 9)
(%i13) factor(1/(n-10)^2+1/(n+10)^2);
                                     2
                                 2 (n  + 100)
(%o13)                        -------------------
                                      2         2
                              (n - 10)  (n + 10)
極めて規則的ですね。合計を求めてみましょう。

(%i16) factor(%o5+%o6+%o7+%o8+%o9+%o10+%o11+%o12+%o13+%o14);
               38         36            34              32                30
(%o16) (2 (10 n   - 6545 n   + 1948485 n   - 349556680 n   + 42208228832 n
                  28                    26                      24
 - 3629088208830 n   + 229250210636238 n   - 10821039069848700 n
                       22                         20
 + 384301849466796030 n   - 10252784171127811225 n
                          18                           16
 + 203306189906202064325 n   - 2926798325861629476180 n
                            14                             12
 + 29146338391530894867520 n   - 178024988388235514261120 n
                             10                              8
 + 363467130642469469054208 n   + 3635885415245379411041280 n
                               6                                4
 - 35075509839165389516955648 n  + 137467668390930878128128000 n
                                2
 - 270063769784651873648640000 n  + 268731604650484039680000000))
          2        2        2        2        2        2        2        2
/((n - 10)  (n - 9)  (n - 8)  (n - 7)  (n - 6)  (n - 5)  (n - 4)  (n - 3)
        2        2        2        2        2        2        2        2
 (n - 2)  (n - 1)  (n + 1)  (n + 2)  (n + 3)  (n + 4)  (n + 5)  (n + 6)
        2        2        2         2
 (n + 7)  (n + 8)  (n + 9)  (n + 10) )

また、ｋ番目は、

(%i14) factor(1/(n-k)^2+1/(n+k)^2);
                                      2    2
                                  2 (n  + k )
(%o14)                         -----------------
                                      2        2
                               (n - k)  (n + k)
ここで、(n^2+k^2)/(n+k)^2は、(n^2+k^2)/(n^2+k^2+2nk)＜１。

第５節　中心から対称な両端項の和

また、最初と最後で折り返してみましょう。

(%i1) factor(1/1^2+1/((2*n+1)-0)^2);
                                    2
                              2 (2 n  + 2 n + 1)
(%o1)                         ------------------
                                           2
                                  (2 n + 1)
(%i2) factor(1/2^2+1/((2*n+1)-1)^2);
                                     2
                                    n  + 1
(%o2)                               ------
                                        2
                                     4 n
(%i3) factor(1/3^2+1/((2*n+1)-2)^2);
                                    2
                              2 (2 n  - 2 n + 5)
(%o3)                         ------------------
                                            2
                                 9 (2 n - 1)
(%i4) factor(1/4^2+1/((2*n+1)-3)^2);
                                  2
                                 n  - 2 n + 5
(%o4)                            ------------
                                           2
                                 16 (n - 1)
(%i5) factor(1/5^2+1/((2*n+1)-4)^2);
                                    2
                              2 (2 n  - 6 n + 17)
(%o5)                         -------------------
                                             2
                                 25 (2 n - 3)
(%i6) factor(1/6^2+1/((2*n+1)-5)^2);
                                  2
                                 n  - 4 n + 13
(%o6)                            -------------
                                            2
                                  36 (n - 2)
(%i7) factor(1/7^2+1/((2*n+1)-6)^2);
                                   2
                             2 (2 n  - 10 n + 37)
(%o7)                        --------------------
                                            2
                                49 (2 n - 5)
(%i8) factor(1/8^2+1/((2*n+1)-7)^2);
                                  2
                                 n  - 6 n + 25
(%o8)                            -------------
                                            2
                                  64 (n - 3)
(%i9) factor(1/9^2+1/((2*n+1)-8)^2);
                                   2
                             2 (2 n  - 14 n + 65)
(%o9)                        --------------------
                                            2
                                81 (2 n - 7)
(%i10) factor(1/10^2+1/((2*n+1)-9)^2);
                                  2
                                 n  - 8 n + 41
(%o10)                           -------------
                                            2
                                 100 (n - 4)

合計をとってみましょう。

(%i11) factor(%o1+%o2+%o3+%o4+%o5+%o6+%o7+%o8+%o9+%o10);


                    20                19                  18
(%o11) (2015568896 n   - 70544911360 n   + 1134489447680 n
                   17                   16                    15
 - 11107425423360 n   + 73939521315456 n   - 353874836595840 n
                     14                     13                     12
 + 1255178716931360 n   - 3349932845936320 n   + 6754430450387156 n
                      11                      10                     9
 - 10224213642506220 n   + 11407876427898225 n   - 9057918591917580 n
                     8                     7                    6
 + 4822732666429886 n  - 1630432406929900 n  + 542389275524825 n
                    5                    4                    3
 - 479700463421340 n  + 392053619272356 n  - 190727139667680 n
                   2
 + 60102887060160 n  - 13855404326400 n + 2016379008000)
                 2        2        2        2  2          2          2
/(1270080 (n - 4)  (n - 3)  (n - 2)  (n - 1)  n  (2 n - 7)  (2 n - 5)
          2          2          2
 (2 n - 3)  (2 n - 1)  (2 n + 1) )


それでは、数値化してみましょう。n=20として、

(%i12) n:20;

(%o12)                                20
(%i13) factor(1/1^2+1/((2*n+1)-0)^2);
                                         2
                                     2 29
(%o13)                               -----
                                        2
                                      41
(%i14) factor(1/2^2+1/((2*n+1)-1)^2);
                                      401
(%o14)                               -----
                                      6  2
                                     2  5
(%i15) factor(1/3^2+1/((2*n+1)-2)^2);
                                    2 5 17
(%o15)                              ------
                                     2   2
                                    3  13
(%i16) factor(1/4^2+1/((2*n+1)-3)^2);
                                     5 73
(%o16)                              ------
                                     4   2
                                    2  19
(%i17) factor(1/5^2+1/((2*n+1)-4)^2);
                                    2 17 41
(%o17)                              -------
                                     2   2
                                    5  37
(%i18) factor(1/6^2+1/((2*n+1)-5)^2);
                                      37
(%o18)                               -----
                                      4  4
                                     2  3
(%i19) factor(1/7^2+1/((2*n+1)-6)^2);
                                     2 13
(%o19)                               -----
                                      2  2
                                     5  7
(%i20) factor(1/8^2+1/((2*n+1)-7)^2);
                                     5 61
(%o20)                              ------
                                     6   2
                                    2  17
(%i21) factor(1/9^2+1/((2*n+1)-8)^2);
                                    2 5 13
(%o21)                              ------
                                     4   2
                                    3  11
(%i22) factor(1/10^2+1/((2*n+1)-9)^2);
                                     281
(%o22)                              ------
                                     10  2
                                    2   5

合計は

(%i23) factor(%o13+%o14+%o15+%o16+%o17+%o18+%o19+%o20+%o21+%o22);


                           776915781552787112787017
(%o23)                     ------------------------
                           498849275011064539161600

小数化すると、

(%i24) float(%o23);

(%o24)                         1.557415877843172
(%i25)

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