
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 18th, 2010, 09:04 PM  #1 
Senior Member Joined: Dec 2006 Posts: 167 Thanks: 3  sequence of the average of a sequence
We know that and What one can say reversely? 
May 18th, 2010, 11:11 PM  #2  
Member Joined: Nov 2007 Posts: 50 Thanks: 0  Re: sequence of the average of a sequence Quote:
A simple counterexample . The convergence of arithmetic means is called Cesaro convergence  perhaps if you google for this term you could find some resources about this topic. E.g. http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation  
May 19th, 2010, 05:09 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: sequence of the average of a sequence
Rewriting, you said: "A sequence that converges to a real value A is a Cauchy sequence with Cesàro mean A (in the limit)." This can be reversed: "A Cauchy sequence with Cesàro mean A (in the limit) converges to A." since Cauchy sequences are convergent and hence converge to the limit of their Cesàro means. As kompik pointed out, "a sequence with Cesàro mean A (in the limit) converges to A" is false. (Of course "a Cauchy sequence converges to A" is false also.) 
May 19th, 2010, 03:24 PM  #4  
Senior Member Joined: Dec 2006 Posts: 167 Thanks: 3  Re: sequence of the average of a sequence Quote:
 
May 19th, 2010, 03:29 PM  #5 
Senior Member Joined: Dec 2006 Posts: 167 Thanks: 3  Re: sequence of the average of a sequence
My Questions is: Prove or disprove that 
May 19th, 2010, 04:21 PM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: sequence of the average of a sequence
Did you miss my post?

May 19th, 2010, 09:18 PM  #7  
Senior Member Joined: Dec 2006 Posts: 167 Thanks: 3  Re: sequence of the average of a sequence Quote:
 
May 20th, 2010, 12:45 AM  #8  
Member Joined: Nov 2007 Posts: 50 Thanks: 0  Re: sequence of the average of a sequence Quote:
. I did not try to prove this rigorously. (Perhaps someone will come up with a simpler counterexample.) But as soon as you notice this equation expressing the sum of some consequent values of a_n: , you can see that in the Cesaro mean you get something like , which converges to zero. Aside note: Theorems like this (when Cesaro convergence + some additional conditions imply the usual convergence) are called Tauberian theorems and they are studied in summability theory. See, for instance, [Korevaar] Tauberian theory\ p.12,Theorem 6.1 [Montgomery Vaughan] Multiplicative Number Theory I\ p.157,Theorem 5.6 [Boos] Classical and Modern Methods in Summability Section 4.1 However, in this case they are formulated for series rather then for limits. (Cesaro summability of a series means that the Cesaro mean of partial sums converges.) Therefore we have to reformulate them slightly to our situation. The above results in our situation say that convergence of Cesaro mean implies convergence if some of the following conditions hold: for each n; for ; for each n; (A is some fixed constant). There are also some results on slowly oscillating series in Boos; however, this is completely new to me, so it would take some time for me to read this.  
May 23rd, 2010, 12:23 PM  #9 
Senior Member Joined: Dec 2006 Posts: 167 Thanks: 3  Re: sequence of the average of a sequence
Thanks a lot kompik. Great counterexample.


Tags 
average, sequence 
Search tags for this page 
average of a sequence converges,sequence average,convergence of cesaro averages,converse of cesaro means,sequence of averages converges to the limit,cesaro mean cauchy,Cesaro's theorem related to average sequence of a sequence,averages of partial sums cauchy,sequence of averages converges
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
sequence  alexmath  Calculus  3  May 16th, 2013 11:55 AM 
Sequence  johnr  Number Theory  7  March 17th, 2013 05:02 PM 
sequence  Algebra  3  April 21st, 2012 07:32 PM  
What's next in this sequence?  yield  Algebra  0  July 26th, 2009 08:05 AM 
cauchy sequence that isn't a fast cauchy sequence  babyRudin  Real Analysis  6  October 10th, 2008 11:11 AM 