By Andrew Hodges on Monday, August 06,
2001 - 03:43 pm:
what is the proof showing that:
1/1+1/4+1/9+1/16+¼ = p2/6?
Can that method of proof be used to determine the sum of any
series like the above, with a higher (even) power, e.g.
1/1 + 1/16 + 1/81..........
By Arun Iyer on Tuesday, August 07, 2001 -
07:51 pm:
Andrew,
for 1,
you will find an answer to this here
For your second question,
these series form a special class of series arising from the
Riemann Zeta function, though the proof of the above question may
not be as useful.
love arun
By Brad Rodgers on Wednesday, August 08,
2001 - 05:15 am:
For the question about
1+1/16+1/81+...
known as z(4), I think Euler proved that this is equal to p4/90
using Bernoulli numbers, I'll try to find a page on this.
Brad
By Andrew Hodges on Wednesday, August 08,
2001 - 12:14 pm:
Can anybody notice any pattern at all in this sequence?
6, 90, 945, 9450, 93555
Andrew
By Brad Rodgers on Friday, August 10, 2001
- 09:45 pm:
The pattern involves Bernoulli numbers, and information about
Bernoulli numbers and the pattern (near the bottom) can be found
at this site
(Incidentally, the pattern is the denominators of z(2k), the numerators
being p2k)
I'm not sure that the Bernoulli numbers could be expressed as
some polynomial, and if so, I have no idea how, but the website
gives a couple of ways to find Bernoulli numbers.
I'm still working on a proof for the relation given between the
Zeta function and Bernoulli numbers.
Brad
By Arun Iyer on Friday, August 10, 2001 -
09:57 pm:
Ah!! Interesting..
So now you can write the given pattern to be...
2(2k)!/4k |B2k |
Am i right??
love arun
By Andrew Hodges on Saturday, August 11,
2001 - 08:07 pm:
Yes, I thought they were the denomiators of z(2k). I calculated them
empirically on my calculator. Bernoulli numbers sound interesting. I'll take
a closer look at those thanks!