### Related

##### product

product(1-a, i, 1, 3);

Calculate

##### product-ratsimp-sum

a(n):=product(n+k,k,0...

a(3);

s(n):=sum(a(k),k,1,n);

Calculate

##### product

product (i, i, 0, 500...

Calculate

##### product-sum

sum(sum((product(k+1+...

Calculate

##### product

eq1: product(e**2, i,...

Calculate

##### product

product(k,k,1, n);

2*4;

Calculate

##### product

product(((n^2+2*n-4)/...

Calculate

##### product

product(k^(n-k),k,1, 3);

2*4;

Calculate

##### product

dens: K(d, f(d, t));

foo: product(dens/(t_...

Calculate

##### product

product(e**2,k,1, n);

2*4;

Calculate

### product

Run Example
```(%i1)I: matrix([1,0],[0,1]);
[ 1  0 ]
(%o1)                              [      ]
[ 0  1 ]
(%i2) S_z: matrix([1,0],[0,-1]);
[ 1   0  ]
(%o2)                             [        ]
[ 0  - 1 ]
(%i3) S_x: matrix([0,1],[1,0]);
[ 0  1 ]
(%o3)                              [      ]
[ 1  0 ]
(%i4) S_y: matrix([0,-%i],[%i,0]);
[ 0   - %i ]
(%o4)                            [          ]
[ %i   0   ]
(%i5) [vals,vecs]:eigenvectors(kronecker_product(S_x,S_y));
(%o5) [[[- 1, 1], [2, 2]], [[[1, 0, 0, - %i], [0, 1, %i, 0]],
[[1, 0, 0, %i], [0, 1, - %i, 0]]]]
(%i6) ```
Run Example
```a(n):=product(n+k,k,0,10)/11!;
product(n + k, k, 0, 10)
(%o1)                  a(n) := ------------------------
11!
(%i2) simpsum:true;
(%o2)                                true
(%i3) s(n):=sum(a(k),k,1,n);
(%o3)                     s(n) := sum(a(k), k, 1, n)
(%i4) factor(s(n));
(%o4) (n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8)
(n + 9) (n + 10) (n + 11))/479001600
(%i5) g(n):=12^(1/3)*(23100*(s(n)+10))^(1/12)-11/2;
1/3                    1/12   11
(%o5)             g(n) := 12    (23100 (s(n) + 10))     - --
2
(%i6) expand(g(n));
12         11       10         9
1/6  1/6   1/3    1/12         n          n        n       11 n
(%o6) 2    5    12    231     expt(--------- + ------- + ------ + ------
479001600   7257600   248832   161280
8         7           6           5           4           3
10831 n    1903 n    242537 n    139381 n    341747 n    190553 n
+ -------- + ------- + --------- + --------- + --------- + ---------
14515200   345600     8709120     1451520     1555200     604800
2
83711 n    n        1     11
+ -------- + -- + 10, --) - --
332640    12       12    2
(%i7) h(n):=n-g(n);
(%o7)                          h(n) := n - g(n)
(%i8) diff(h(n),n);
11        10        9        7        5        3
24 n   + 132 n   + 220 n  - 264 n  + 264 n  - 132 n  + 20 n
(%o8) 1 - (-----------------------------------------------------------
24
10         9         8         6         4        2
330 n   + 1650 n  + 2475 n  - 2310 n  + 1650 n  - 495 n  + 25         9
+ ------------------------------------------------------------- + 1320 n
6
8           7           6           4          2
54450 n  + 217800 n  + 254100 n  - 127050 n  + 36300 n  - 1815         8
+ -------------------------------------------------------------- + 5940 n
3
7            6            5            3
1262184 n  + 4417644 n  + 4417644 n  - 1472548 n  + 210364 n         7
+ ------------------------------------------------------------ + 7920 n
8
6            5            4           2
1804110 n  + 5412330 n  + 4510275 n  - 902055 n  + 42955
+ --------------------------------------------------------
2
5             4             3
20501580 n  + 51253950 n  + 34169300 n  - 3416930 n         5
+ --------------------------------------------------- - 5544 n
6
4             3             2
25228500 n  + 50457000 n  + 25228500 n  - 840950             3             2
+ ------------------------------------------------ + 12756216 n  + 29759004 n
3
12       11       10       8       6       4       2       11        10        9        7        5        3                          9          8          7          5          3                             8           7           6           4           2                    7           6           5           3                      6             5            4            2                     5             4            3
11/6  11/6   11/3    11/12       2 n   + 12 n   + 22 n   - 33 n  + 44 n  - 33 n  + 10 n    30 n   + 165 n   + 275 n  - 330 n  + 330 n  - 165 n  + 25 n        10   6050 n  + 27225 n  + 36300 n  - 25410 n  + 12100 n  - 1815 n        9   157773 n  + 631092 n  + 736274 n  - 368137 n  + 105182 n         8   257730 n  + 902055 n  + 902055 n  - 300685 n  + 42955 n   3416930 n  + 10250790 n  + 8542325 n  - 1708465 n         6   5045700 n  + 12614250 n  + 8409500 n  - 840950 n            4            3             2
+ 20633832 n + 3585840)/(2     5     12     231      expt((------------------------------------------------------- + ----------------------------------------------------------- + 132 n   + ------------------------------------------------------------ + 660 n  + --------------------------------------------------------- + 990 n  + ------------------------------------------------------- + -------------------------------------------------- - 924 n  + ------------------------------------------------ + 3189054 n  + 9919668 n  + 10316916 n  + 3585840 n)/39916800 + 10,
24                                                           6                                                                       3                                                                      8                                                                   2                                                      6                                                            3
11
--))
12
(%i9) factor(479001600);
10  5  2
(%o9)                           2   3  5  7 11
(%i10) factor(11!);
8  4  2
(%o10)                           2  3  5  7 11
(%i11) ```
Run Example
```load(draw);
(%o1)            /usr/share/maxima/5.21.1/share/draw/draw.lisp
(%i2) coord: setify(makelist(k,k,-4,4));
(%o2)                 {- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4}
(%i3) points2d: listify(cartesian_product (coord ,coord ));
(%o3) [[- 4, - 4], [- 4, - 3], [- 4, - 2], [- 4, - 1], [- 4, 0], [- 4, 1],
[- 4, 2], [- 4, 3], [- 4, 4], [- 3, - 4], [- 3, - 3], [- 3, - 2], [- 3, - 1],
[- 3, 0], [- 3, 1], [- 3, 2], [- 3, 3], [- 3, 4], [- 2, - 4], [- 2, - 3],
[- 2, - 2], [- 2, - 1], [- 2, 0], [- 2, 1], [- 2, 2], [- 2, 3], [- 2, 4],
[- 1, - 4], [- 1, - 3], [- 1, - 2], [- 1, - 1], [- 1, 0], [- 1, 1], [- 1, 2],
[- 1, 3], [- 1, 4], [0, - 4], [0, - 3], [0, - 2], [0, - 1], [0, 0], [0, 1],
[0, 2], [0, 3], [0, 4], [1, - 4], [1, - 3], [1, - 2], [1, - 1], [1, 0],
[1, 1], [1, 2], [1, 3], [1, 4], [2, - 4], [2, - 3], [2, - 2], [2, - 1],
[2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [3, - 4], [3, - 3], [3, - 2],
[3, - 1], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [4, - 4], [4, - 3],
[4, - 2], [4, - 1], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4]]
(%i4) vf2d(x,y):=  vector ([x,y],[cos(y),x]/6);
[cos(y), x]
(%o4)              vf2d(x, y) := vector([x, y], -----------)
6
(%i5) vect2: makelist(vf2d(k[1],k[2]),k,points2d );
cos(4)    2                        cos(3)    2
(%o5) [vector([- 4, - 4], [------, - -]), vector([- 4, - 3], [------, - -]),
6       3                          6       3
cos(2)    2                        cos(1)    2
vector([- 4, - 2], [------, - -]), vector([- 4, - 1], [------, - -]),
6       3                          6       3
1    2                      cos(1)    2
vector([- 4, 0], [-, - -]), vector([- 4, 1], [------, - -]),
6    3                        6       3
cos(2)    2                      cos(3)    2
vector([- 4, 2], [------, - -]), vector([- 4, 3], [------, - -]),
6       3                        6       3
cos(4)    2                        cos(4)    1
vector([- 4, 4], [------, - -]), vector([- 3, - 4], [------, - -]),
6       3                          6       2
cos(3)    1                        cos(2)    1
vector([- 3, - 3], [------, - -]), vector([- 3, - 2], [------, - -]),
6       2                          6       2
cos(1)    1                      1    1
vector([- 3, - 1], [------, - -]), vector([- 3, 0], [-, - -]),
6       2                      6    2
cos(1)    1                      cos(2)    1
vector([- 3, 1], [------, - -]), vector([- 3, 2], [------, - -]),
6       2                        6       2
cos(3)    1                      cos(4)    1
vector([- 3, 3], [------, - -]), vector([- 3, 4], [------, - -]),
6       2                        6       2
cos(4)    1                        cos(3)    1
vector([- 2, - 4], [------, - -]), vector([- 2, - 3], [------, - -]),
6       3                          6       3
cos(2)    1                        cos(1)    1
vector([- 2, - 2], [------, - -]), vector([- 2, - 1], [------, - -]),
6       3                          6       3
1    1                      cos(1)    1
vector([- 2, 0], [-, - -]), vector([- 2, 1], [------, - -]),
6    3                        6       3
cos(2)    1                      cos(3)    1
vector([- 2, 2], [------, - -]), vector([- 2, 3], [------, - -]),
6       3                        6       3
cos(4)    1                        cos(4)    1
vector([- 2, 4], [------, - -]), vector([- 1, - 4], [------, - -]),
6       3                          6       6
cos(3)    1                        cos(2)    1
vector([- 1, - 3], [------, - -]), vector([- 1, - 2], [------, - -]),
6       6                          6       6
cos(1)    1                      1    1
vector([- 1, - 1], [------, - -]), vector([- 1, 0], [-, - -]),
6       6                      6    6
cos(1)    1                      cos(2)    1
vector([- 1, 1], [------, - -]), vector([- 1, 2], [------, - -]),
6       6                        6       6
cos(3)    1                      cos(4)    1
vector([- 1, 3], [------, - -]), vector([- 1, 4], [------, - -]),
6       6                        6       6
cos(4)                         cos(3)
vector([0, - 4], [------, 0]), vector([0, - 3], [------, 0]),
6                              6
cos(2)                         cos(1)
vector([0, - 2], [------, 0]), vector([0, - 1], [------, 0]),
6                              6
1                       cos(1)
vector([0, 0], [-, 0]), vector([0, 1], [------, 0]),
6                         6
cos(2)                       cos(3)
vector([0, 2], [------, 0]), vector([0, 3], [------, 0]),
6                            6
cos(4)                         cos(4)  1
vector([0, 4], [------, 0]), vector([1, - 4], [------, -]),
6                              6     6
cos(3)  1                      cos(2)  1
vector([1, - 3], [------, -]), vector([1, - 2], [------, -]),
6     6                        6     6
cos(1)  1                    1  1
vector([1, - 1], [------, -]), vector([1, 0], [-, -]),
6     6                    6  6
cos(1)  1                    cos(2)  1
vector([1, 1], [------, -]), vector([1, 2], [------, -]),
6     6                      6     6
cos(3)  1                    cos(4)  1
vector([1, 3], [------, -]), vector([1, 4], [------, -]),
6     6                      6     6
cos(4)  1                      cos(3)  1
vector([2, - 4], [------, -]), vector([2, - 3], [------, -]),
6     3                        6     3
cos(2)  1                      cos(1)  1
vector([2, - 2], [------, -]), vector([2, - 1], [------, -]),
6     3                        6     3
1  1                    cos(1)  1
vector([2, 0], [-, -]), vector([2, 1], [------, -]),
6  3                      6     3
cos(2)  1                    cos(3)  1
vector([2, 2], [------, -]), vector([2, 3], [------, -]),
6     3                      6     3
cos(4)  1                      cos(4)  1
vector([2, 4], [------, -]), vector([3, - 4], [------, -]),
6     3                        6     2
cos(3)  1                      cos(2)  1
vector([3, - 3], [------, -]), vector([3, - 2], [------, -]),
6     2                        6     2
cos(1)  1                    1  1
vector([3, - 1], [------, -]), vector([3, 0], [-, -]),
6     2                    6  2
cos(1)  1                    cos(2)  1
vector([3, 1], [------, -]), vector([3, 2], [------, -]),
6     2                      6     2
cos(3)  1                    cos(4)  1
vector([3, 3], [------, -]), vector([3, 4], [------, -]),
6     2                      6     2
cos(4)  2                      cos(3)  2
vector([4, - 4], [------, -]), vector([4, - 3], [------, -]),
6     3                        6     3
cos(2)  2                      cos(1)  2
vector([4, - 2], [------, -]), vector([4, - 1], [------, -]),
6     3                        6     3
1  2                    cos(1)  2
vector([4, 0], [-, -]), vector([4, 1], [------, -]),
6  3                      6     3
cos(2)  2                    cos(3)  2
vector([4, 2], [------, -]), vector([4, 3], [------, -]),
6     3                      6     3
cos(4)  2
vector([4, 4], [------, -])]
6     3
(%i6) apply(draw2d , append ([ color=blue], vect2 ));
(%o6) [gr2d(vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector, vector, vector, vector, vector, vector, vector, vector, vector,
vector)]
(%i7) ```

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