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Forum index » Science and Technology » Math » Recreational
Euler Two
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Jim Dars
science forum beginner


Joined: 03 Jul 2005
Posts: 41

PostPosted: Tue Jun 20, 2006 8:21 pm    Post subject: Euler Two Reply with quote

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
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