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Some Critical Points of the Hyperpower Function y = xxx . .


Much has been written about the arithmetic and geometric sequences, {nx} and {xn}. Here some of the properties of another sequence, the hyperpowers {nx}. These are studied in the interval (0,1 ) using the derivatives and the analytical tools such as L'Hopital's rule and Newton's method. The sequence {nx} has two subsequences (green) odd {Ox}and (red) even{Ex} depending on whether n is even "E" or odd "O". The former has one relative minimum point (as does the generalized parabola {xE}). The other subsequence on the other hand {nx} has a number of points of inflection, but no relative minimum. The analytical methods are too cumbersome without the aid of computers which explains why so little work has been done on these functions in the past. Eighteen different properties are mentioned and they can all be verified by students who are skilled in the uses of computers. Due to the unpredictable compatibility of different web sites, many of the intricate formulas are omitted, but can be found in my article "Some Critical Points of the Hyperpower Function xxx . " by J. F. MacDonnell found in the International Journal of Mathematical Education in Science and Technology , 1989, Vol. 20, #2 PP> 297-305.
The interval of convergence for the
HYPERPOWER function y = xxx . .

Introduction to them infinitely iterated exponential function y = f(x) = xxx . .

Just as addition generalizes to multiplication so multiplication generalizes to exponentiation. Each operation generates its own sequence, the last one involves such tedious calculations it has been out of reach until the computer age. This study attempts to gather some information about these interesting functions which arise from a string of finite or infinite length of exponentiations.
By way of introduction consider the following problem. The equation with the infinitely iterated exponentiation
y = f(x) = xxx . . = 2 can be solved by the algebraic maneuver
xxx . . = y = 2 then x y also = 2 so that x2= 2 and then x equals the square root of 2.
It would seem then that x would equal the cube root of 3 when solving the equation y = 3, but this is not the case. In fact, this procedure will only work when
e-1 < y < e so that 1/(ee) < y < e(1/e)
This has been established by Knoebel [I] and Mitchelmore [2] in their treatments of iterated exponentials. Knoebel introduces the notation nx for these 'hyperpowers' where x, xx, xxx ., . . are
written1x,2x, 3x . . . and y in the example above would be the limit of the sequence y = lim nx.
as n approaches infinity. He then proves that the sequence {nx} converges only in the interval
e-e < x < e1/e
. The members of this sequence of hyperpowers offer an interesting study in critical points and calculus teachers seeking computer-oriented problems which would challenge their brighter students may find the sequence useful. This is an attempt to organize the tools to help a teacher present this sequence {nx}.


Iterated exponentials have been puzzling people for centuries, reaching back to mathematicians such as Daniel Bernouli who discussed it with Christian Goldbach back in 1728 in an exchange of letters. Fascinated also have been Georg Cantor, Arthur Cayley, Leonard Euler, Ferdinand Eisenstein, Ludwig Seidel and Ernst Schroeder. Most recently R. Arthur Knoebel, providing a bibliography with no less than 125 entries, has summarized the findings using especially the more recent work and terminology of Daihachiro Sato. Euler, in fact was the first to prove that the function y = f(x) = y = xxx . . converges for all x in the interval [e-e, e1/e] and diverges for any other positive value of x.
At the other end of the interval in (1, e1/e) the sequence {nx} continues to converge, but for x > e1/e it diveges. There is an inflection point at x = .582 . . This is found by using a computer or calculator to find the zero in equation (8) by ploting its graph of (8) and noting the change of signs of y" on either side of x = .582 . .

Three sequences of hyperpowers

Like the sequence of powers of x, x2, x3, x4, . . . = {xn} the sequence of hyperpowers {nx} divides naturally into two subsequences depending on whether n is odd {Ox} or even {Ex} where O represents the set of odd integers and E the set of even integers. Members of {Ox} have no extrema while members of {Ex} have one minimum point. After proving that the sequence {nx} converges for all x in the interval (e-e , e1/e) Knoebel establishes the convergence of these two subsequences in the interval (0, e1/e).
The functions in these sequences happen to be particularly well suited for computer analysis and without this aid, computation of the derivatives would be too tedious. This study examines the graphs of these functions and is divided into four parts.
1 The value of y = lim nx as n approaches infinity for x in the intrval (0, 1] shown in figure 1
2 Members of {Ex} for x in (0,1) shown in figure 2;
3 Members of {Ox} for x in (0,1) shown in figure 3 and
4 Enlargement around the bifurcation point *(e-e, e-1)

To accomplish these calculations, the first and second derivatives must be established and then recursion formulas programmed in the computer. It is important to use the proper convention of reading the powers downward, so that:
ln ( 333) = (33) ln3 or 27 ln 3 AND NOT 3 ln (33)
Therefore ln (xxx) = (xx) lnx AND NOT x ln (xx)
Knoebel established not only that this function y = xxx . . is continuous and differentiable in the interval but also that since it is always increasing in the interval, it has an inverse so that dy/dx= 1 /(dx/dy). To study this limit function, it is first necessary to compute the first and second derivatives. This infinitely iterated exponential converges in the interval (e-e, e1/e). This function which is the limit of the sequence of functions x, 2x, 3x, 4x . . . nx. . . y = {nx} is continuous and has derivatives in this interval. The derivatives y' and y" are found by elementary calculus methods.
The recursion formulas for the first derivative nx' and second derivative nx" for values of n = 2, 3, 4, . . . can - by logarithmic differentiation and mathematical induction - be proven to be:

[1] nx' = nx [n-1x' lnx + (n-1x)/x] and
[2] nx" = nx'[n-1x' lnx+ (n-1x)/x] + nx[n-1x" lnx + {2x(n-1x') - (n-1x)}/x2]

The first few derivatives are;

[3] 2x' = 2x (lnx + 1)
[4] 3x' = 3x [2x' lnx + (2x)/x]

[5] 2x" = 2x'(lnx+ 1) + 2x[lnx + 1)2 = 1/x]
[6] 3x" = 3x'[2x' lnx+ (2x)/x] + 3x[2x" lnx + {2x(2x' )-(2x)}/x2]
Now consider this infintiely iterated expontential function y = lim xxx . . which converges in the interval
(e-e , e 1/e). The function, which is the limit of the sequence of functions x, 2x, 3x, 4x, . .nx . . . = {nx} is continuoius and has derivatives in this interval. The derivatives y' and y" are found by logarithmic differention after noticing that
xy = y or
x = y1/yx so that lnx = lny/y and
(dx/dy)/x = - lny/(y2) + 1/y2 so that
dx/dy = [y1/y(1-lny)]/y2 then y' = dy/dx = 1/[dx/dy] and the first derivative is

[7] y' (y2)/[y1/y(1-lny)]
The second derivative is obtained by differentating y' and with some algebra becomes
[8] y" = [y'/y]2[{y + 2y(1-lny)-(1-lny)2}/(1-lny)]
These eight formulae are sufficient to compose a computer program to find the derivatives nx' and nx"
Figure 1
Points on the graphs of certain hyperpowers using values of n = 1, 2, 3, 4, . . . O, E, O, E, . . . in the interval (0,1). The sequence of functions {nx} converges as n-> only in the interval [e-e, e1/e] where e-1 < y < e. At the bifurcation point *(e-e, e-1) the sequence {nx} divides into two (green) odd {Ox} and (red) even{Ex} sequences each of which converges in the interval (0, e-e)

Part 1 y = lim nx (n->infinity) only in the interval for x in [e-e, e1/e] shown in Figure 1

The graph of the function y can be seen in figure 1. Since the derivative is defined only on the open interval, it cannot be used to find the minimum point at x = e-e, but several other things can be proven.

Figure 2
Points on the graphs of certain even hyperpowers using values of E = 2, 4, 10, 40, 80 in the interval (0,1): i.e. 2x, 4x, 10x, 40x, 80x. Each curve has a minimum point (m) which moves leftward to the point (e-e, e-1). Inflection points are grouped together at symbol .

Part 2 Even subsequence {ex} for x in (0, 1) shown in Figure 2

Use of these computer programs on the subsequence of {nx} choosing n to be an even integer E; 2x, 4x, 6x, ... Ex. The following seven observations can be made.
(I) In (0,1) 2x is always concave up and there is a minimum point at x = e-1.
(2) As x->0 the limit of 2x can be found to be 1.
(3) The slope at x="0" is vertical.
(4) At x ="1", 2x(l)= 1 and the slope is 1 and the second derivative is 2.
(5) Computation of the minimum point for the members of the subsequence when n>3 is too difficult without computer trial and error.
(6) It is easy to see that 2x" > 0 for the whole interval so there are no inflection points on the 2x curve. Not as evident is that Ex" > 0 for the whole interval for E < 8. But two inflection points appear for all members when E < 10 and four inflection points appear for all members when E < 16. While it would be hopelessly tedious to try to compute where these second derivatives become zero, the computer use of the recursion formulas shows positive and negative values over the interval (0, 1) and affords an opportunity to pinpoint these inflection points as was done in the process for finding the minimum point.
(7) From figure 2, several interesting facts are evident. The subsequence is nested in the interval (0,1). That is 2x > 4x > 6x > 8x . . . And the minimum point m moves to the left as e increases towards the value x = e-e = .065988 as seen in figure 4.

Figure 3
Points on the graphs of certain odd hyperpowers using values of O = 1, 3, 5, 9, 17, 41, 81 in the interval (0,1). Each curve has two inflection points: the left inflection point is identified by symbol O and the right uses the positive symbol +. The maximum value of the function (Ox - x) is displayed at point M. As O -> infinity M approaches the point (e-e, e-1).

Part 3 Odd subsequence {Ox} for x in (0, 1) shown in Figure 2

When the first and second derivative formulas and the computer programs are applied to the subsequence of odd values of n, 1x, 3x, 5x, 7x, 9x . . . Ox the following nine observations can be made.

(I) The first member of {Ox} where the odd number O is chosen to be I is simply the y = x line.
(2) Apart from the case O = 1 the odd computations for 3x are more difficult to obtain.
(3) The fact that 3x(0)->0 can be obtained from the results of 2x(0) above.
(4) As n->infinity Lim3x'(x)->1
(5) That there are no extrema for the odd case n = O.
(6) Where the second derivative changes sign inflection points appear visible in figure 3.
(7) Although 3x has no maximum, an interesting point (M) does occur which is where the maximum distance 3x is from the y = x line: i.e. at M f(x) = (3x - x) has a maximum value.
(8) For the rest of the members of this subsequence we find two inflection points, no extrema and a maximum value at x = M for the function f(x) = Ox - x as seen in figures 3 and 4.
(9) From figure 3, it is noticed that the subsequence { x} is nested: x < 3x < 5x < 7x < . . . in the interval (0,1 ) and point M moves to the left as O gets larger.

Figure 4
Enlargement of the bifurcation point *(e-e, e-1) where the sequence {nx} divides into two odd {Ox} and even {Ex} sequences each of which converges in the interval (0, e-e]. Visible also are a four minima (m) and two Maximum (M) values already described. They both approach the bifurcation point *(e-e,e-1) which is approximately (.06598, .36788).

Part 4 The function y = lim xxx . . where x is in the interval
(e-e, e 1/e) and its bifurcation point *(e-e, e-1) shown in Figure 4

The graph of the function y is seen in figure 1. Since the derivative is defined only on the open interval, it cannot be used to find the minimum point at x = e-e, but two other facts can be proven. (I) One inflection point I is found in this interval using the process described above. It is found that y" (.3944) = 0 and concavity changes on either side of this point (.3944, .5819).
(2) Newton's method of approximation can be used to find the maximum distance M from the y = 1x line. These two points, I and M and the two end points are enough to give us the shape of the graph seen in figures 3 and 4.

Comparison of the three sequences

Just as the computer has greatly facilitated the computation of the derivatives to study these three sequences, so also the computer can be used to check the values in the interval and outside. So if x = 2 this is in the interval repeated exponentiation and will give the answer y = 21/2. But for x = 3 we get y = 2.4780 and not 31/3. This verifies an earlier claim. In fact for a value close to zero, for example, x = .01 we find that for after a large number of exponentiations
O(.01) = 0.0130925 and E(.01) = .9414883.
Also (.01).9414883 = .0l30925 and (.01).0130925 = .9414883.

This illustrates that the sequence {n.01 } is diverging (by oscillation) but the two subsequences {O(.01)} and {E(.01)} are converging to two distinct limits .0130925 and .9414883.


1. Carmichael, R. D., On certain transcendental functions.1908, Mathematical Monthly, Vol. 15,p. 78.
2. Goebel, F.; Nederpelt, R.P. The number of numerical outcomes of iterated powers. 1971, Mathematical Monthly, Vol.78, p. 1097.
3. Hurwitz, Solomon On the rational solutions of mn = nm with n not equal to m. 1967,Mathematical Monthly, Vol. 74, p. 298.
4. Laidler, P.; Landau, B. V. A power sequence exercise for a pocket calculator. 1977, Math Gazette, Vol. 61, p. 191.
5. Knoebel, R. A. Exponential reiterated. 1981, Mathematical Monthly, Vol. 88, p.235.
6. MacDonnell, J. F. Some critical points of the hyperpower function xxx . . International Journal of Mathematical Education, 1989, Vol. 20, #2, p.297.
7. Mitchelmore, M. C., A matter of definition. 1974, Mathematical Monthly, Vol. 81, p. 643.
8. Moulton, E. J. The real function defined by yx = xy. 1916, Mathematical Monthly, Vol. 23, p. 233.

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