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January 28th, 2011, 10:21 AM   #1
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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.
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January 28th, 2011, 10:51 AM   #2
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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.
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January 28th, 2011, 02:54 PM   #3
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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 ...
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February 3rd, 2011, 10:25 AM   #4
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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.
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February 3rd, 2011, 12:03 PM   #5
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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...
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February 3rd, 2011, 01:18 PM   #6
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Re: Convergent series -> series of geometric means converges

It must be true since it's the contrapositive of:
If then converges
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February 3rd, 2011, 01:46 PM   #7
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Re: Convergent series -> series of geometric means converges

But that statement isn't true either! Isn't the harmonic series a counterexample?
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February 3rd, 2011, 02:42 PM   #8
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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.
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February 6th, 2011, 10:07 AM   #9
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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
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February 6th, 2011, 10:18 AM   #10
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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
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