
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
June 9th, 2017, 07:31 AM  #1 
Member Joined: May 2017 From: France Posts: 50 Thanks: 1  Convergent series : cos(k^2)/k
Hi, $\displaystyle \textbf{Convergent series? : } \text{the series } \sum \limits_{k=0}^n \frac{\cos(k^2)}{k} \text{ is it convergent?}$ Cordially. Last edited by skipjack; June 9th, 2017 at 09:24 AM. 
June 9th, 2017, 07:45 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,854 Thanks: 2228 Math Focus: Mainly analysis and algebra 
Apparently so (0.444336 ish), but I have no idea how one would prove it. https://www.wolframalpha.com/input/?...D1+to+infinity 
June 9th, 2017, 08:16 AM  #3 
Member Joined: May 2017 From: France Posts: 50 Thanks: 1 
There is a tip.

June 9th, 2017, 09:26 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87 
If cosk^2 > a for infinitely many k the series diverges because sum 1/k diverges.
Last edited by skipjack; June 9th, 2017 at 09:01 PM. 
June 9th, 2017, 09:51 AM  #5 
Member Joined: May 2017 From: France Posts: 50 Thanks: 1 
I don't think. Moreover by using the same reasoning as you, we would have that the sum of cos (k) / k also diverges. 
June 9th, 2017, 12:55 PM  #6 
Global Moderator Joined: May 2007 Posts: 6,255 Thanks: 507 
These series have + and  terms, so it is hard to prove or disprove convergence.

June 9th, 2017, 12:59 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,854 Thanks: 2228 Math Focus: Mainly analysis and algebra  
June 9th, 2017, 01:02 PM  #8 
Math Team Joined: Jul 2011 From: Texas Posts: 2,569 Thanks: 1272  
June 9th, 2017, 02:42 PM  #9  
Member Joined: May 2017 From: France Posts: 50 Thanks: 1  Quote:
So I keep looking, and I throw the thread as soon as I find one, if someone finds one, before he can put it here. Bye.  
June 19th, 2017, 06:28 AM  #10 
Member Joined: May 2017 From: France Posts: 50 Thanks: 1  

Tags 
convergent, cosk2 or k, serie, series 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Convergent Series  Dmath  Calculus  2  June 4th, 2014 06:22 AM 
Is this series convergent?  nappysnake  Real Analysis  1  December 18th, 2011 07:49 PM 
series convergent of divergent  aaronmath  Calculus  8  November 6th, 2011 10:15 PM 
series convergent or divergent  aaronmath  Calculus  2  November 6th, 2011 09:14 PM 
Convergent series > series of geometric means converges  The Chaz  Real Analysis  11  February 7th, 2011 05:52 AM 