wu :: forums
« wu :: forums - SIGMAarctan(2/n^2) »
Welcome, Guest. Please Login or Register.
Nov 28th, 2024, 6:06am

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   riddles
   putnam exam (pure math) (Moderators: towr, Icarus, william wu, Grimbal, Eigenray, SMQ)
   SIGMAarctan(2/n^2)		 « Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: SIGMAarctan(2/n^2)  (Read 1653 times)
ThudnBlunder
Uberpuzzler
*****




The dewdrop slides into the shining Sea

   


Gender: male
 Posts: 4489
SIGMAarctan(2/n^2)  
« on: May 20th, 2008, 5:37am »		 Quote Quote Modify 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: male
 Posts: 2276
Re: SIGMAarctan(2/n^2)  
« Reply #1 on: May 20th, 2008, 10:44am »		 Quote Quote Modify Modify
135o
IP Logged
ThudnBlunder
Uberpuzzler
*****




The dewdrop slides into the shining Sea

   


Gender: male
 Posts: 4489
Re: SIGMAarctan(2/n^2)  
« Reply #2 on: May 20th, 2008, 10:52am »		 Quote Quote Modify Modify
on May 20th, 2008, 10:44am, Barukh wrote:
135o

Was that computer-assisted?   Roll Eyes
IP Logged
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
Barukh
Uberpuzzler
*****






   


Gender: male
 Posts: 2276
Re: SIGMAarctan(2/n^2)  
« Reply #3 on: May 20th, 2008, 11:19am »		 Quote Quote Modify Modify
on May 20th, 2008, 10:52am, ThudanBlunder wrote:

Was that computer-assisted?   Roll Eyes

 
No.
IP Logged
ThudnBlunder
Uberpuzzler
*****




The dewdrop slides into the shining Sea

   


Gender: male
 Posts: 4489
Re: SIGMAarctan(2/n^2)  
« Reply #4 on: May 20th, 2008, 11:27am »		 Quote Quote Modify Modify
on May 20th, 2008, 11:19am, Barukh wrote:

 
No.

Then I'm beginning to believe our literary tastes are similar.   Tongue
IP Logged
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
Barukh
Uberpuzzler
*****






   


Gender: male
 Posts: 2276
Re: SIGMAarctan(2/n^2)  
« Reply #5 on: May 20th, 2008, 11:14pm »		 Quote Quote Modify 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: male
 Posts: 1948
Re: SIGMAarctan(2/n^2)  
« Reply #6 on: May 21st, 2008, 2:45am »		 Quote Quote Modify 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
*****





   
WWW

Gender: male
 Posts: 1291
Re: SIGMAarctan(2/n^2)  
« Reply #7 on: May 21st, 2008, 3:59am »		 Quote Quote Modify 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 Roll Eyes
« Last Edit: May 21st, 2008, 4:00am by william wu » IP Logged

[ wu ] : http://wuriddles.com / http://forums.wuriddles.com
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board