From: james.jones@hal9k.ann-arbor.mi.us (James Jones) 
Newsgroups: sci.math
Subject: Closed forms for sums
Date: 8 Jan 93 00:36:00 GMT
Distribution: world
Organization: HAL 9000 BBS, W-NET HQ, Ann Arbor, Michigan, USA
Reply-To: james.jones@hal9k.ann-arbor.mi.us (James Jones) 

OB>Subject: sum(i=1 to n, i^3) = ??   URGENT !!

While we're on the subject, I was killing some time with Derive, a
Mathematical Assistant one day waiting form my Finite class to begin
and found the closed form for

        ---
        \     j
         >   i
        /
        ---
        i=1

where j went from 1 to about 12 or 13.  Can't remember exactly where I
stopped at.

Noticed some interesting patterns that developed there.

Some things I remember:

1.      All closed forms had  n  and  (n + 1)  as a factors.

                            2              2
2.      All odd j > 1 had  n   and  (n + 1)   as a factors.


3.      Add even j > 2 had interesting coefficients on the terms in
        the last factor.
        (at least the last factor the way Derive factored it)

j = 4

                3      2
  n (n + 1) (6 n  + 9 n  + n - 1) 
 ---------------------------------
                 30               

j = 6

               5       4      3      2
 n (n + 1) (6 n  + 15 n  + 6 n  - 6 n  - n + 1) 
------------------------------------------------
                       42                       

j = 8

                7       6       5       4       3       2
 n (n + 1) (10 n  + 35 n  + 25 n  - 25 n  - 17 n  + 17 n  + 3 n - 3)
---------------------------------------------------------------------
                                  90

See the pattern?

when j = 4, the last pair of coefficients are:
        1,-1
when j = 6, the last two pair of coefficients are:
        6,-6  and  -1,1
when j = 8, the last three pair of coefficients are:
        25,-25  -17,17  and  3,-3

not shown, but ...

when j = 10, the last four pair of coefficients are:
        28,-28  -38,38  28,-28  and  -5,5
when j = 12, the last five pair of coefficients are:
        1575,-1575  -3430,3430  5150,-5150  -3859,3859  and 691,-691

Derive returns all of these in a very short time.  I don't think that
Derive has programmed in all of the closed forms (it may, but I doubt
it), so my question is, how does one go about determining a closed
form for a sum?

I plan on having my College Algebra class this spring use derive and
graphing calculators to help with some of the work, but I would like
something I could use as a discovery exercise for them.  Hopefully
something to do with discovering closed forms.  Mathematical Induction
always gives the College Algebra students trouble and I hate to just
say:

        ---
        \      2      n  ( n + 1 )  ( 2n + 1 )
Prove:   >    i   =   -------------------------
        /                        6
        ---
        i=1

and would like them to be able to see where we came up with those
formulas.

That is, if its not too hard.  I tend to push more on my students than
the other instructors, so I need to make sure it is not beyond their
level.

Any explanation or references (explanation preferred) would be
appreciated.
---
 . DeLuxe./386 1.25 #1158s . A Parallelogram is a figure that looks like this.
 . QNet3. . [WNet] 2/3 Board  Decatur, IL  217-877-1138  USR DS V32/V42b
                                   
----
+------------------------------------------+-----------------------------+
| HAL 9000 BBS +1 313 663 4173 or 663 3959 | Four 14.4k v.32bis dial-ins |
| Public Access QWK-to-Usenet gateway      | With PCBoard 14.5aM & uuPCB |
+------------------------------------------+-----------------------------+