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January 28th, 2011, 10:21 AM  #1 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Convergent series > series of geometric means converges
(feel free to edit this and convert my statements to latex. I'm bad at latex, and on my phone...) Claim: if the infinite series a_n converges, and b_n := geometric mean of the first n terms of a, then the infinite series b_n converges. This was "left as an exercise", and my first impulse was to use AMGM, but the inequality goes in the wrong direction! I was thinking about just taking the square root of each, but they aren't necessarily positive... Besides "use the definitions", I'd appreciate any hints. 
January 28th, 2011, 10:51 AM  #2 
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: Convergent series > series of geometric means converges
Isn't that "sequence", not "series"? I can't see how (b_n) could be a series.

January 28th, 2011, 02:54 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Convergent series > series of geometric means converges
If you add them all up, then they are a series! b1 = a1 b2 = sqrt(a1*a2) b3 = cuberoot(a1*a2*a3) b4 = ... Sum from n=1 to infinity of b_n converges if sum from n = 1 to infinity of a_n converges. Certainly we could express these statements in terms of the sequences an and bn, but that doesn't suggest a solution ... 
February 3rd, 2011, 10:25 AM  #4 
Member Joined: Jan 2011 Posts: 36 Thanks: 0  Re: Convergent series > series of geometric means converges
Perhaps proving the contrapositive, namely "If the sum of b_n diverges then the sum of a_n diverges", might be easier? Just fiddling around with it If diverges then But Then we may write Eh, I have to go now. But my intuition is telling me that this might be easier to prove. Will try again later. 
February 3rd, 2011, 12:03 PM  #5  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Convergent series > series of geometric means converges Quote:
 
February 3rd, 2011, 01:18 PM  #6 
Member Joined: Jan 2011 Posts: 36 Thanks: 0  Re: Convergent series > series of geometric means converges
It must be true since it's the contrapositive of: If then converges 
February 3rd, 2011, 01:46 PM  #7 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Convergent series > series of geometric means converges
But that statement isn't true either! Isn't the harmonic series a counterexample?

February 3rd, 2011, 02:42 PM  #8  
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: Convergent series > series of geometric means converges Quote:
 
February 6th, 2011, 10:07 AM  #9 
Member Joined: Jan 2011 Posts: 36 Thanks: 0  Re: Convergent series > series of geometric means converges
My mistake. Just to be clear. Is this statement equivalent to the question? If for then for where 
February 6th, 2011, 10:18 AM  #10  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Convergent series > series of geometric means converges Quote:
 

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