 My Math Forum Convergent series -> series of geometric means converges
 User Name Remember Me? Password

 Real Analysis Real Analysis Math Forum

 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 AM-GM, 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: 938 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:
 Originally Posted by GeminiDreams ... If diverges then ...
I think the converse is true, but not this statement... 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: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Convergent series -> series of geometric means converges

Quote:
 Originally Posted by GeminiDreams It must be true since it's the contrapositive of: If then converges
That statement, like its contrapositive, is false. The harmonic series and the sum of the reciprocals of the primes are the most famous counterexamples. 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:
 Originally Posted by GeminiDreams My mistake. Just to be clear. Is this statement equivalent to the question? If for then for where
That, except with a lowercase "i" as the indexing subscript of a Tags >, convergent, converges, geometric, means, series ,

# prove that the geometric mean of a convergent sequence converges

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post fgp Real Analysis 1 June 27th, 2010 09:36 PM coolaid317 Calculus 1 April 23rd, 2010 04:23 AM nedaiii Real Analysis 1 February 8th, 2009 07:15 PM cindyyo Algebra 2 August 24th, 2008 01:25 AM clooneyisagenius Real Analysis 1 January 30th, 2008 12:50 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      