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Exponential Difficulties (Posted on 2004-10-30) Difficulty: 4 of 5
R = N ^ (N ^ (N ^ ...)). What is the maximum N>0 that will yield a finite R?

No Solution Yet Submitted by SilverKnight    
Rating: 3.0000 (1 votes)

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Solution Solution | Comment 2 of 6 |
Since R=N^R, we need to find the largest N such that the curves f(R)=R and g(R)=N^R have a point of intersection for some N>0.  For small N, these curves intersect at two points, while for large N, they don't intersect at all.  The maximum N is such that they intersect at exactly one point. 

At this point, the curves are tangent; since f has slope 1, we have g'(R)=1, or ln(N)*N^R = 1.  Plugging in R=N^R gives R*ln(N)=1.  But we can also take the ln of both sides to get ln(ln(N))+R*ln(N)=0, so that ln(ln(N))=-1.  The maximal N is thus e^(1/e).

  Posted by David Shin on 2004-10-30 17:32:55
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