Project 18: Repeated Exponents
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The number need not equal . For example, , so , but , or . "Exponentiation is not associative."

The limit


as the number of exponentiations tends to infinity is an interesting one. This can be viewed as a discrete dynamical system, (as in CD text Chapter 20).

  • Let g[x]=bx and define a sequence a0=1, a1=g[a0], ..., an+1=g[an]. Compute a1, a2, a3, a4, and a5 explicitly.

    In function notation, the sequence is repeated computation of the function, a0=1, a1=g[a0]=g[1], a2=g[a1]=gg[g[1]], a3=g[a2]=g[g[g[1]]], ...

    In CD Chapter 20 of the main text, these "discrete dynamical systems" are studied in some detail. The program FirstDynSys from CD Chapter 20 can be used to compute and graph the sequence. Try it for b=1.2, b=0.8 and some other values. (Make the function g[p]=bp and f[p]=g[p]-p. Then a1=b1, a2=bb, a3=ba2=bbb, .)

    If the limit


    equals , then we must have and .

  • Show that the equation has no solutions if b=3. Start with a plot,

    Plot[{3x, x},{ x, 0, 2}]

  • Show graphically that the equation has solutions if b=1.1 or if b<1.


    : b<1        and    The Biggest Solution Base
    We would like to know the value of b where y=bx just touches y=x. Larger values of b give no solutions to x=bx.

  • The Biggest b
    Show that is equivalent to the equation x=bx.

    Use the computer to graph .

    Prove that the maximum of is 1/e.

    Use what you have so far to prove that is the largest b so that x=bx has a solution.

    Run FirstDynSys with b=e(1/e).

    Theorem CD 20.4 of the main text gives a condition for the sequence g[g[g[...]]] to converge (when we start close enough to the limit). It simply says we must have |g'[xe]|<1 where g[xe]=xe. Now we want to know the value of the base b<1 that crosses y=x at slope -1.

  • The Lower Limit
    Calculate the derivative g'[x] and show that Theorem CD 20.4 says we must have

    Solve the pair of equations with the hints:

    The first equation gives .

    This means xe=1/e. Why?

    Substitute xe=1/e into the first equation to show that b=1/ee.

    Calculate using the computer.

    Run FirstDynSys with b=0.065988 for 1000 terms of the sequence.

    Run FirstDynSys with b=0.06 for 1000 terms of the sequence.

    There is much more on this problem in the American Math Monthly, vol. 88, nr. 4, 1981, in the article, Exponentials Reiterated, by R. Arthur Knoebel.

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