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simpsum

Run Example
(%i1)sum((-1^n)/(2^n), n, 0, inf), simpsum;
(%o1)                                 - 2
(%i2) 
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
f:1/(1+x);
                                       1
(%o1)                                -----
                                     x + 1
(%i2) s(n,x):=integrate((-x)^n,x,0,1);
                                               n
(%o2)                s(n, x) := integrate((- x) , x, 0, 1)
(%i3) s(0,x);
(%o3)                                  1
(%i4) s(1,x);
                                        1
(%o4)                                 - -
                                        2
(%i5) s(2,x);
                                       1
(%o5)                                  -
                                       3
(%i6) s(3,x);
                                        1
(%o6)                                 - -
                                        4
(%i7) g3:sum((-x)^k,k,0,3);
                                  3    2
(%o7)                          - x  + x  - x + 1
(%i8) g:sum((-x)^k,k,0,n-1),simpsum;
                                       n
                                  (- x)  - 1
(%o8)                             ----------
                                   - x - 1
(%i9) S3:integrate(g3,x,0,1);
                                      7
(%o9)                                 --
                                      12
(%i10) fmg:f-g;
                                           n
                                1     (- x)  - 1
(%o10)                        ----- - ----------
                              x + 1    - x - 1
(%i11) fmg:factor(fmg);
                                         n
                                    (- x)
(%o11)                              ------
                                    x + 1
(%i12) 
[abs,plot2d,simpsum,sin,sum] [additive,declare,inf,simpsum] [allroots,factor,inf,simpsum,sum] [assume,binomial,simpsum] [assume,inf,simpsum] [augcoefmatrix,coefmatrix,col,load,matrix,simpsum,sum] [binomial,cos,factor,load,next_prime,plot2d,prev_prime,simpsum,sin,sqrt,sum,true] [binomial,cos,factor,load,next_prime,prev_prime,simpsum,sin,sqrt,sum,true] [binomial,simpsum,subst,sum] [binomial,simpsum,sum,true] [binomial,simpsum,sum] [binomial,simpsum,true] [binomial,simpsum] [cauchysum,false,gensumnum,inf,intosum,psexpand,simpsum,sumcontract,sumexpand,true] [cauchysum,false,gensumnum,inf,intosum,simpsum,sumcontract,sumexpand,true] [cauchysum,false,intosum,simpsum,sum,sumcontract,sumexpand,true] [cos,diff,matrix,plot2d,plot3d,simpsum,sin,true] [cos,inf,simpsum,sum] [cos,simpsum,sum,taylor] [cos,simpsum,sum] [debugmode,simpsum,sum,true] [declare,ev,inf,limit,simpsum,sum] [diff,expand,factor,product,simpsum,true] [diff,matrix,simpsum,sin,true] [diff,simpsum,sum] [ev,makelist,simpsum,sum] [expand,factor,product,simpsum,true] [expand,simpsum,sum] [factor,inf,makelist,numer,powerdisp,simpsum,sum,sumcontract,true] [factor,inf,makelist,numer,powerdisp,simpsum,sum,true] [factor,integrate,simpsum,sum] [factor,simpsum,sum] [factorsum,simpsum,sum] [false,simpsum] [fullratsimp,inf,integrate,limit,simpsum,true] [inf,integrate,simpsum,sum] [inf,linsolve,simpsum,sum] [inf,makelist,numer,powerdisp,simpsum,sum,true] [inf,simpsum,sum] [inf,simpsum,true] [inf,simpsum] [intosum,simpsum,sum,sumcontract] [simpsum,sqrt,sum,true] [simpsum,sqrt,sum] [simpsum,subst,sum] [simpsum,sum,sumcontract] [simpsum,sum,true] [simpsum,sum] [simpsum,true] [simpsum]

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