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*From*: John Mallinckrodt <ajmallinckro@CSUPomona.Edu>*Date*: Thu, 27 Feb 1997 09:19:34 -0800 (PST)

And one last word from me on the rubber worm...

Aha! Uri's result for the discrete case points me to the proof of

the result that I had noticed numerically. As he found, the

fraction of the rope covered after step n in the discrete case is

f = (sum from k=1 to n of (1/k)) / C

where C is the single parameter for the problem which I defined as

(rope speed/worm speed). Combining this with my result that the

time (analogous to n) required to cover a fraction of the rope in

the continuous case is

(t = ) n_cont = e^(fC) - 1

we get the ratio of time (or number of steps) required for a given

f in the continuous case to that in the discrete case is

r = n_cont/n

= (e^(fC) - 1)/n

= [e^(sum from k=1 to n of (1/k)) - 1]/n

A ratio that is independent of C as I had noticed numerically.

Now I was interested in the limit of this ratio

r_inf = limit as n -> infinity of r

= lim as n -> inf of e^(sum from k=1 to n of (1/k))/n

Taking natural logs of both sides

ln(r_inf) = lim as n -> inf of [(sum from k=1 to n of (1/k)) - ln(n)]

which is one of the known representations of Euler's constant, gamma.

Thus,

r_inf = exp(gamma)

QED

Finally, I can get back to work!

John

----------------------------------------------------------------

A. John Mallinckrodt email: mallinckrodt@csupomona.edu

Professor of Physics voice: 909-869-4054

Cal Poly Pomona fax: 909-869-5090

Pomona, CA 91768 office: Building 8, Room 223

web: http://www.sci.csupomona.edu/~mallinckrodt/

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