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Consider the infinitely nested exponential equation
xxx⋅⋅⋅=N.\large x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = N.xxx⋅⋅⋅=N.
One might naively say, "Easy, just substitute in," and
xN=N, so x=NN.x^N = N, \ \text{ so }\ x = \sqrt[N]{N}.xN=N, so x=NN.
However, this doesn't converge for all NNN. What is the highest NNN for which it does?
Give your answer to 3 decimal places.
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