robison@uiucdcsb.UUCP (11/02/84)
ln (v+1)
Given: f = --------
v
f(n) = nth composition of f
Find: limit f(n) as n approaches infinity for v>1
(I restate the problem to so as clarify my interpretation. I hope it
is correct.)
If the limit exists, then it is a solution of:
ln (v+1)
v = --------
v
which can be rearranged to the problem of finding the root to:
2
v - ln (v+1) = 0
The numerical solution is 0.7468817423085, I don't know what the "symbolic" is.
To show there is a limit, just find an interval [A,B] and number M such that:
f([A,B]) is a subset of [A,B]
|df/dv| < M < 1 for any v on [A,B]
One such interval is [.5,2.8]. If we start with v > 2.8, it is easy to
show that the repeated composition of f(f(v)) converges to a number < 2.8,
and thus on the interval [.5,2.8].
- Arch Robison @ uiucdcs
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