Author |
Topic: SIGMAarctan(2/n^2) (Read 1654 times) |
|
ThudnBlunder
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
SIGMAarctan(2/n^2)
« on: May 20th, 2008, 5:37am » |
Quote Modify
|
Evaluate tan-1(2/n2) n=1
|
|
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: SIGMAarctan(2/n^2)
« Reply #1 on: May 20th, 2008, 10:44am » |
Quote Modify
|
135o
|
|
IP Logged |
|
|
|
ThudnBlunder
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: SIGMAarctan(2/n^2)
« Reply #2 on: May 20th, 2008, 10:52am » |
Quote Modify
|
on May 20th, 2008, 10:44am, Barukh wrote: Was that computer-assisted?
|
|
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: SIGMAarctan(2/n^2)
« Reply #3 on: May 20th, 2008, 11:19am » |
Quote Modify
|
on May 20th, 2008, 10:52am, ThudanBlunder wrote: Was that computer-assisted? |
| No.
|
|
IP Logged |
|
|
|
ThudnBlunder
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: SIGMAarctan(2/n^2)
« Reply #4 on: May 20th, 2008, 11:27am » |
Quote Modify
|
on May 20th, 2008, 11:19am, Barukh wrote: Then I'm beginning to believe our literary tastes are similar.
|
|
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
|
Re: SIGMAarctan(2/n^2)
« Reply #5 on: May 20th, 2008, 11:14pm » |
Quote Modify
|
hidden: | Solution is based on the following identity: tan-1(2/n2) = tan-1(n+1) - tan-1(n-1) |
|
|
IP Logged |
|
|
|
Eigenray
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 1948
|
|
Re: SIGMAarctan(2/n^2)
« Reply #6 on: May 21st, 2008, 2:45am » |
Quote Modify
|
Or less cleverly, working out the first few partial sums suggests hidden: | arctan{-(n-1)(n+2)/[n(n-3)]} + arctan{2/n2} = arctan{-n(n+3)/[(n+1)(n-2)]} |
|
|
IP Logged |
|
|
|
william wu
wu::riddles Administrator
Gender:
Posts: 1291
|
|
Re: SIGMAarctan(2/n^2)
« Reply #7 on: May 21st, 2008, 3:59am » |
Quote Modify
|
Digression: As a knee jerk reaction, I took the derivative of the summand, and tried summing that instead. Not that that would lead to anything relevant for this problem ... but I ended up with something that surprised me: d/dx [ArcTan[2/x^2]] = -(4 x)/(4 + x^4) -(4 n)/(4 + n^4) = -3/2 OK, now someone explain why I shouldn't be surprised
|
« Last Edit: May 21st, 2008, 4:00am by william wu » |
IP Logged |
[ wu ] : http://wuriddles.com / http://forums.wuriddles.com
|
|
|
|