gulf [38]% maple
    |\^/|     Maple V Release 2 (University of Victoria)
._|\|   |/|_. Copyright (c) 1981-1992 by the University of Waterloo.
 \  MAPLE  /  All rights reserved. MAPLE is a registered trademark of
 <____ ____>  Waterloo Maple Software.
      |       Type ? for help.
> sum( k*k, k=1..n );
 
                              3              2
                   1/3 (n + 1)  - 1/2 (n + 1)  + 1/6 n + 1/6
 
> sum( k^2, k=1..n );
 
                              3              2
                   1/3 (n + 1)  - 1/2 (n + 1)  + 1/6 n + 1/6
 
> simplify( " );
 
                                 3        2
                            1/3 n  + 1/2 n  + 1/6 n
 
> factor( " );
 
                            1/6 n (n + 1) (2 n + 1)
 
> factor( sum( k^2, k=1..n ) );
 
                            1/6 n (n + 1) (2 n + 1)
 
> sum( 1/k, k=1..n );
 
                               Psi(n + 1) + gamma
 
> sum( harmonic(k), k=0..n-1 );
 
                               n - 1
                               -----
                                \
                                 )   harmonic(k)
                                /
                               -----
                               k = 0
 
> harmonic(3);
 
                                  harmonic(3)
 
> help(harmonic);
   
FUNCTION: harmonic - The Harmonic Function
   
CALLING SEQUENCE:
   harmonic(x)
   
PARAMETERS:
   x - an expression
   
SYNOPSIS:   
- The harmonic function is defined as follows:
   
      harmonic(n) = sum( 1/i, i=1..n ) = Psi(n+1) + gamma
   
EXAMPLES:   
> harmonic(1);
                                  harmonic(1)
   
> evalf(");
                                       1.
   
> harmonic(2);
                                  harmonic(2)

> evalf(");
                                  1.500000000
   
SEE ALSO:  inifcns
>       

> simplify( sum( harmonic(j), j=0..n-1 ) );
 
                               n - 1
                               -----
                                \
                                 )   harmonic(j)
                                /
                               -----
                               j = 0
 
> simplify( sum( sum( 1/(k-j), j=1..k-1 ), k=1..n ) );
 
                                n   /k - 1      \
                              ----- |-----      |
                               \    | \      1  |
                                )   |  )   -----|
                               /    | /    k - j|
                              ----- |-----      |
                              k = 1 \j = 1      /
 
> sum( k*x^k, k=0..n );
 
                   (n + 1)
                  x        ((n + 1) x - n - 1 - x)       x
                  -------------------------------- + --------
                                     2                      2
                              (x - 1)                (x - 1)
 
> simplify( " );
 
                        (n + 2)      (n + 1)      (n + 1)
                     - x        n + x        n + x        - x
                   - ----------------------------------------
                                            2
                                     (x - 1)