The counterexamples

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Here are the three counterexamples to the conjecture below. I don't want to write anything about them so I just post the original reply. Again, my sincerest thanks to the three for making my life simpler.

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From: "James Yorke"
Date: Tue, 22 Feb 2005 14:17:02 -0500
Try 10 to the power - n squared for c(n).

From: "Jose Perico Esguerra"
Date: Wed, 23 Feb 2005 13:11:15 +0000 (GMT)
Counterexample: c(n)=0.9^(n^3)

From: "Richard Taylor"
Date: Wed, 23 Feb 2005 15:13:52 -0500
I think this is not true. For example try c(n)=e(n)/2^n where e(n) is 1 if n has remainder 0 or 1 when divided by 3 and where e(n) is -1 if n has remainder 2 when divided by 3. If I didn't make a mistake this gives d(n)=-6 if 3 divides n, d(n)=2/3 if n has remainder 1 when divided by n and d(n)=-1 if n has remainder 2 when divided by n. This sequence does not converge to anything.

Best wishes,
Richard Taylor