The Repeated Power Function
The Repeated Power Function
Leonhard Euler proved that the function (x)=, where appears times, approaches a limit for <x< as . (The red points indicate the limits at the endpoints of the interval of convergence.) This Demonstration shows the values of (x) for to . You can zoom in to see the limits at and . The expression bifurcates at the lower endpoint () with ()1/e. The sequence (x) converges to 1 as from above while (x) converges to 0. Beyond the upper endpoint () the expression rapidly grows to infinity.
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1/e
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n∞
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x=
1/e
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x=

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2n
x0
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2n+1
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1/e
e