Summation and Products

 

In Maple

For comparison with other CAS choose from: Axiom Derive Macsyma Mathematica Reduce

inputs outputs

    sum( m^2 * x^m, m=1..n );

          (n + 1)               2                  2              2  2
         x        (- 2 (n + 1) x  + 2 (n + 1) x + x  + x + (n + 1)  x

                         2            2    /        3   x (x + 1)
              - 2 (n + 1)  x + (n + 1) )  /  (x - 1)  - ---------
                                         /                      3
                                                         (x - 1)

    sum( cos( (2*m-1)* Pi / (2*n+1) ), m=1..r );

           Pi (r + 1)              Pi   2     Pi (r + 1)
     - cos(----------) (- 2 cos(-------)  sin(----------)
             2 n + 1            2 n + 1         2 n + 1

                     Pi        Pi (r + 1)         Pi          Pi (r + 1)
          + 2 cos(-------) cos(----------) sin(-------) + sin(----------))
                  2 n + 1        2 n + 1       2 n + 1          2 n + 1

            /        Pi             Pi
           /  sin(-------) + cos(-------)
          /       2 n + 1        2 n + 1

    simplify( " );

             Pi (r + 1)         Pi   2     Pi (r + 1)
      (2 cos(----------) cos(-------)  sin(----------)
               2 n + 1       2 n + 1         2 n + 1

                      Pi        Pi (r + 1) 2        Pi
           - 2 cos(-------) cos(----------)  sin(-------)
                   2 n + 1        2 n + 1        2 n + 1

                 Pi (r + 1)      Pi (r + 1)           Pi           Pi
           - cos(----------) sin(----------) + cos(-------) sin(-------))
                   2 n + 1         2 n + 1         2 n + 1      2 n + 1

             /        Pi
            /  sin(-------)
           /       2 n + 1

    product( exp( sin(m*x) ), m=1..n );

    (exp(

          - sin((n + 1) x) cos(x) + sin((n + 1) x) + cos((n + 1) x) sin(x)
      1/2 ----------------------------------------------------------------
                                     cos(x) - 1

           /           sin(x)
      ))  /  exp(1/2 ----------)
         /           cos(x) - 1