I was preparing a "collage" (of "worm problem" messages for my home page)
and could not find my last contribution to this thread. Perhaps it was
deleted from my e-mail archive; perhaps I did post it.
So I am posting it below. Where is the error? Two proofs (one based on
the cumulative algorithm and one based on iterative algorithm) lead to
mutually exclusive conclusions. One of them MUST have an error. I am doing
what we always do in calculus; ignoring an infinitesimally small term in
a sum of very large terms.
Ludwik Kowalski
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I now understand the nuances of the worm problem and I feel good about
this. But there is a little cloud in the blue sky. My mathematical analysis
shows that the bug will never catch the tractor. I know it contradicts
what has been accepted. It also contradict the "physics argument" which
can be stated as follows. The tractor has no acceleration, the bug has
a finite positive acceleration. Therefore, sooner or later, the bug will
catch the tractor, no matter how large the difference between the
initial speeds. That is how I would now answer the first question in
Leigh's problem.
Clearly my "proof" (that n --> infinity) must have a hidden fault. But I do
not see it. Please help me find the error in what is written below. Note
that I am using curly brackets { } for subscripts. Do not confuse {n+1},
which is a subsript, with (n+1), which is a place holder for a value. Yes,
is is easier on paper but we must communcate with ASCII. My x=0 is where
the bug is at t=0.
.......................................................................
At time t=n the tractor's location (along the rigid x axis) is
L{n}=10^5+10^5*n=10^5*(n+1) (A)
The location of the bug, at t=n, is given by iterative formula
x{n}=[x{n-1}+1]*(1+1/n) (B)
That formula yields exactly the same values as the non-iterative formula o
which contains a sum of 1/k terms (Uri's notation). As everybody else, I
say that the bug will catch the tractor when L{n}=x{n}. At that moment
x{n}-x{n-1}=LST=10^5 (C)
where LST is the length of the last step. (the last speed of the bug
is 10^5 cm/s, the same as the speed of the tractor). I know, from the
sollutions of Chip, Uri or John, that x{n} and x{n-1} are gigantic in
comparison with 10^5. What is one kilometer (10^5 cm) in comparison with
zilions of trip around the universe? Therefore I can assume, while writing
L{n}=x{n}, that x{n}=x{n-1}. Once I do this I conclude that 1/n --> 0 and
n --> infinity, as shown below. What is wrong?
L{n}=[x{n-1}+1]*(1+1/n) or (D)
L{n}=[L{n}-LST]*(1+1/n) or (E)
L{n}=[L{n}-0]*(1+1/n) (F)
I know that n=10^43429.443 and n--> infinity are practically identical.
But conceptually they are very different. How can a neglection of an
unsignificant term, LST, change a finite solution of a problem into
infinity? Where did I err?