Another counterexample

00000110

Here's another counterexample to the (wrong) conjecture. This time it's from Marty Golubitsky from the University of Houston.

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Date: Tue, 8 Mar 2005 23:31:51 -0600 (CST)
From: "Marty Golubitsky"

I believe that the conjecture is false.

A colleague mentioned this counterexample.

Let e(n) be the sequence

1, 1/2, -1/3, -1/4, 1/5, 1/6, -1/7, -1/8 , ...

Define c(n) by c(1) = 0 and c(n) = c(n-1) + e(n-1)

Then c(2) = c(1) + e(1) = 1
c(3) = c(2) + e(2) = 1 + 1/2
c(4) = c(3) + e(3) = 1 + 1/2 - 1/3
etc

Claim: c(n) is convergent

c(n) = 1 + (1/2-1/3) - (1/4-1/5) + (1/6-1/7) + ...
=1 + 1/6 - 1/20 + 1/42 - 1/72
=1 + 14/120 + ...

In particular, note that

1/k - 1/(k+1) - 1/(k+2) + 1/(k+3) = (4k+6)/(k(k+1)(k+2)(k+3))
< 4/(k(k+1)(k+2))
< 4/k^3

whose sum converges.

It follows that

d(n) = e(n-1)/e(n)

Then e(n-1)/e(n) = n/(n-1) times 1, -1, 1, -1, 1, -1, ...

which does not converge.