Jim Dars science forum beginner
Joined: 03 Jul 2005
Posts: 41

Posted: Tue Jun 20, 2006 8:21 pm Post subject:
Euler Two



Hi All,
In Euler One I wrote:
***********
I was just rereading William Dunham's "Euler  Master of Us All". Not as
good as his wonderful "Journey Through Genius", but still a VERY interesting
book.
He points out Euler's solution(s) to find the value of
Sum (1 to infinity on k) k^i where i is an even integer (see Euler
Two)
and remarks that no one has yet found solutions for i being an odd integer
greater than one. (Except for the 1978 "find" that for i=3 the answer is
irrational.) Has any progress yet been made?
********************
Let S represent the sum from 1 to infinity on the index j.
Dunham presents Euler's solution to find the sum for even k. It is of the
form
S_j of r_j = A
S_j of (r_j)^2 = A*S_ of r_j  2*B
S_j of (r_j)^3 = A*S_j (r_j)^2 B*S_ of r_j +3*C
Thus he uses A to find the sum for k^2 as (pi^2)/6
and Euler, being Euler, carries out calculation, BY HAND, to show
for k^26 one obtains 1315862*(pi^26)/11094481976030578125
using M.
Now some time ago we had a discussion of what constituted a closed form
solution. Frankly, at the time it didn't peak my interest and I paid little
attention. But now I started wondering. Has Euler created a closed form
solution? What if I need the answer to 11094481976030578126! ??
Comments appreciated [other than my linking two threads together (Euler One
& Euler Two) which I KNOW some don't appreciate.]
Best wishes, Jim 
