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From: robison@uiucdcsb.UUCP
Newsgroups: net.math
Subject: Re: Limit Problem
Message-ID: <9700029@uiucdcsb.UUCP>
Date: Thu, 1-Nov-84 18:31:00 EDT
Article-I.D.: uiucdcsb.9700029
Posted: Thu Nov  1 18:31:00 1984
Date-Received: Sat, 27-Oct-84 04:20:47 EDT
References: <27000002@uiucuxc.UUCP>
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Nf-ID: #R:uiucuxc:27000002:uiucdcsb:9700029:000:886
Nf-From: uiucdcsb!robison    Oct 25 23:31:00 1984


              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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