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 to 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.
.......................................................................
At time t=n the tractor's location 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 the same values as the non-iterative formula of Uri.
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. I know, from the sollution 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
round 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?