TThe strange properties of the infinite power tower
An “investigative math” approach for young students
Luca Moroni ∗ (August 2019)Nevertheless, the fact is that there is nothing as dreamy and poetic,nothing as radical, subversive, and psychedelic, as mathematics.Paul Lockhart – “ A Mathematician’s Lament ” Abstract
In this article we investigate some ”unexpected” properties of the “
Infinite Power Tower ”function (or “ Tetration with infinite height ”): y = f ( x ) = x x xx ...where the “tower” of exponentiations has an infinite height.Apart from following an initial personal curiosity, the material collected here is alsointended as a potential guide for teachers of high-school/undergraduate studentsinterested in planning an activity of “ investigative mathematics in the classroom ”, wherethe knowledge is gained through the active, creative and cooperative use of diversifiedmathematical tools (and some ingenuity).The activity should possibly be carried on with a laboratorial style, with no preclusionson the paths chosen and undertaken by the students and with little or no informationimparted from the teacher’s desk.The teacher should then act just as a guide and a facilitator.The infinite power tower proves to be particularly well suited to this kind of learningactivity, as the student will have to face a challenging function defined through a ratheruncommon infinite recursive process. They’ll then have to find the right strategies to getaround the trickiness of this function and achieve some concrete results, without the helpof pre-defined procedures.The mathematical requisites to follow this path are: functions, properties of exponentialsand logarithms, sequences, limits and derivatives. The topics presented should then beaccessible to undergraduate or “advanced high school” students. keywords — infinite power tower, tetration, fixed-point, recursion, recursive sequence,cobweb, Euler, Lambert, Lagrange ∗ Liceo Scientifico ”Donatelli-Pascal” - Milan - Italy We adopt here the quite popular term power tower even if it is not entirely correct. The expression shouldinstead be described in terms of exponentiations and not powers . a r X i v : . [ m a t h . HO ] A ug ontents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54 The problem of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85 Fixed points and convergence criteria (in general) . . . . . . . . . . . . . . . . . . . .
86 Fixed points and convergence of the power tower (the algebraic route) . . . . . . .
107 Fixed points and convergence of the power tower (the graphical route) . . . . . . .
118 Outside the convergence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189 Some history about the power tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . After presenting the infinite power tower function, its definition and its unexpected properties(section - Introduction ), we start an investigation about its mathematical characteristics. Insection ( Generalization ) we introduce the function y = x y and its inverse function x = y /y that prove to be useful to give some promising clues on the infinite power tower. In section ( The problem of convergence ) we introduce the problem of the convergence of the recursivesequence leading to the infinite power tower. This problem is furtherly investigated in thesections ( Fixed points and convergence criteria (in general) ), ( Fixed points and convergenceof the power tower (the algebraic route )) and ( Fixed points and convergence of the powertower (the graphical route) ), where the investigation is brought forward with both algebraic andgraphical methods. Section ( Outside the convergence interval ) explores what happens outsidethe convergence interval and the emergence of a periodic cycle for the values given by the powertower function. Lastly, in section ( Some history about the power tower ) we briefly discusssome very interesting historical aspects on the origin of the interest about the infinite powertower (where the main characters are Lambert, Euler and Lagrange).2
Introduction
Let’s define the “
Infinite Power Tower ” function (or “
Tetration with infinite heights ”) as: y = f ( x ) = x x xx ...where the tower of exponentiations has an infinite height.People aware of the explosive nature of exponential functions will guess that, if x >
1, the f ( x )previously defined will soon blow up to infinity as the height of the tower is increased. But,contrary to this initial guess, some trial with a pocket calculator suggests that there might be astable behavior for some set of values, even with x > x = √ √ √ √ √ √ .. .. .. → y = √ √ √ √ √ √ ... = y → y = (cid:16) √ (cid:17) y → y y = √ → y = 2We could be tempted to extend and generalize the procedure in the following way y = x (cid:40) x xxxxx... (cid:41) = y → y = x y → y y = x so that, setting y = 3 it would be x = 3 = √ y = 4 it would be x = √ √ x = √ y : 2 or 4? √ √ √ √ √ .. .. .. = √ √ √ √ √ .. .. .. →
2? 4?Let’s check again numerically.After having defined the following recursive function in
Mathematica or in
Geogebra
Mathematica:
PowerTower[a , k Integer] : = Nest[Power[a,
Geogebra:
Iteration(a^x, a, n - 1) (where a and n can be defined as sliders)we find that a tower with height=1000, starting from x = √
2, yields a result of 2 (as expected,anyway not “4”), but if the starting point is √ In the rest of this article we’ll assume that x, y ∈ (cid:60) and x > y > x , there is a definite valuefor the y .Another strange thing happens when we give the x some values close to 0 and consider odd/evennumbers for the height of the tower:It seems that a small change in the height of the tower may produce a relevant change in theresult. How can it be?These initial experiments suggest the following practical questions to be addressed: • Why is f (cid:16) √ (cid:17) = 2 (and not 4)? • Why is not f (cid:16) √ (cid:17) = 3? • For which values of x do we have a definite (finite) value of y ? • Why do we sometimes get two different values whit small changes in the height of thetower? 4
Generalization
We have previously defined the infinite power tower function as: y = f ( x ) = x x xx ...where the tower of exponentiations has an infinite height.Firstly, let’s make clear what is the conventional meaning of applying subsequentexponentiations.In order to do so it’s convenient to start from the definition of the related functions representingtowers with finite heights . It will be: y n = f n ( x ) = x x xx... } n times so it is y = f ( x ) = x, y = f ( x ) = x x , y = f ( x ) = x x x and so on.It’s important to observe that it is x x x = x ( x x ) and not x x x = ( x x ) x = x x .This means that the tower is built from the highest exponent downwards to the lowest level.This example shows the difference between a downwards and an upwards construction:3( ) = 3 = 7 625 597 484 987 (cid:54) = (cid:16) (cid:17) = 9 = 729Using the definition of the “finite” tower f n ( x ), the infinite power tower can then be re-definedas follows: y = f ( x ) = x x xx ... = lim n →∞ f n ( x )Alternatively, we can build the sequence of functions { y , y , y , y , ..., y n , ... } and takeadvantage of the fact that this sequence can be defined recursively as: (cid:40) y = xy n +1 = x y n f ( x ) = lim n →∞ y n It’s easy to check that with above definition we have y = xy = x y = x x y = x y = x x x ... reproducing, when n → ∞ , our infinite power tower.After having clarified the meaning of the infinite power tower function y = f ( x ) we can saythat, if it converges to some finite value y , than it is y = x (cid:40) x xxx ... (cid:41) = y → y = x y The inverse function will then be x = g ( y ) = y y ( g = f − )5nlike y = x y (that’s not the expression in explicit form of a function), this appears to bea well-defined function (although mapping y (cid:55)→ x ) for any value y >
0. So, let’s study thecharacteristics of this function x = g ( y ) = y y to get some insight on the function f ( x ) we aremostly interested in.Note that we’ll use the following useful identity in some calculation: x = y y = e ln y y = e lny / y Expression : x = y y Domain : y > Limits : lim y → y y = 0 + lim y →∞ y y = 1in fact, using the L’Hpital’s rule (H), y /y = e lny/y and lim y → + ln yy = −∞ , lim y → + ∞ ln yy H = 0 First derivative : dxdy = ddy (cid:16) e lny / y (cid:17) = e lny / y (cid:16) − ln yy (cid:17) = y y (cid:16) − ln yy (cid:17) Asymptotes : the line x = 1 is a horizontal asymptote Stationary points : dxdy = 0 → − ln y = 0 → y = e → x = e edxdy > → − ln y > → y < e The point M ( y M ; x M ) = (cid:16) e ; e e (cid:17) is a maximum. Second derivative: d xdy = y y − (1 − y + (2 y + ln y − · ln y ) Inflection points: − y + (2 y + ln y − · ln y = 0 → (cid:40) y F ≈ . , x F ≈ . y F ≈ . , x F ≈ . g ( y ) is: Fig. 1: Plot of x = g ( y ) = y /y We must remember that in the plot above, differently from the usual conventions, the verticalaxis represents the x and the horizontal axis is the y .If we rotate the graph we get, now with the usual orientation of the axis, the set of pointssatisfying the equivalent relations: x = y y or y = x y or y = e y ln x ig. 2: Implicit plot of y − x y = 0 But since g ( y ) is not invertible as it is not bijective, the plot shown in Fig. 2 is not that of afunction.The inverse function of g ( y ) could only be defined on a proper restriction of the domain of g ,where the function is a bijection.For example, this condition would be respected in the region defined by 0 < x ≤ e e ∧ y > Fig. 3: Possible inverse function of g ( y ) The problem of convergence
The plot of Fig. 2 represents all the points satisfying the equation y = x y . Anyway, it wouldbe problematic to say that these points are also the ones satisfying the equation of the infinitepower tower y = f ( x ) = x x xx ...In fact, above equation is written in the form of a function, whilst y = x y is not the expressionof a function.Furthermore the plot tells us that for some values of the x (with 1 < x < e e ) we would gettwo possible values of the y and this doesn’t make much sense with how the f ( x ) is defined.The problem is hidden in the following passage: If the infinite power tower converges than it is y = x x xxxxx... = x (cid:40) x xxxxx... (cid:41) = y → y = x y But the truth is that the infinite power tower doesn’t converge for every values of x .How can we tell that? And how can we find the interval of convergence?We must recall that the function f ( x ) can be defined by recursion as the limit of a sequence offunctions with finite heights: (cid:40) y = xy n +1 = x y n ⇒ f ( x ) = lim n →∞ y n So, given some value of x , we can say that the sequence { y n } converges if it stabilizes to somefinite value as far as n is increased.In practice, the convergence requires that lim n →∞ y n +1 = y n (or lim n →∞ y n +1 − y n = 0) . To find the conditions assuring the convergence of a recursive sequence we abandon temporarilyour power tower function and explore, in more general terms, sequences, fixed points and whena sequence is bound to converge to a fixed point.
In general, given a sequence defined by its starting value y and by the recursion equation y n +1 = r ( y n ), where r is a smooth function, a fixed point y ∗ is a value satisfying the equation y ∗ = r ( y ∗ ). The name fixed point means that if y n = y ∗ then y n +1 = r ( y n ) = r ( y ∗ ) = y ∗ andthe sequence will keep on re-producing the same value for all future iterations.Once we have found the fixed point(s) of a sequence by solving the equation y = r ( y ) we maybe interested to know if a fixed point is stable (or attractive) or not.If the fixed point is attractive then, when we start close to it, we will end up even closer. Inmathematical terms we can say that, calling δ n the distance between y ∗ and y n ( δ n >
0) andstarting from a point y n = y ∗ ± δ n the subsequent term will be y n +1 = y ∗ ± δ n +1 , and therequirement for the convergence is that δ n +1 < δ n ∀ n . Since it is8 n +1 = | y n +1 − y ∗ | = | r ( y n ) − r ( y ∗ ) | and δ n = | y n − y ∗ | we have δ n +1 δ n = | r ( y n ) − r ( y ∗ ) || y n − y ∗ | The closer we are to y ∗ the more above ratio will approximate the absolute value of the derivative | r (cid:48) ( y ∗ ) | . This suggests that it’s possible to use the mean value theorem to state that there exista point ξ ∈ ( y n , y ∗ ) such that | r (cid:48) ( ξ ) | = | r ( y n ) − r ( y ∗ ) || y n − y ∗ | In our case we can say that there exists a point ξ ∈ ( y n , y ∗ ) such that δ n +1 δ n = | r (cid:48) ( ξ ) | Then, if there is some interval in which it is | r (cid:48) ( y ) | < k < ∀ y ∈ ( | y ∗ − y | < δ n ) it will also be δ n +1 δ n = | r (cid:48) ( ξ ) | < k and | r ( y n ) − r ( y ∗ ) | < k | y n − y ∗ | that is | y n +1 − y ∗ | < k | y n − y ∗ | , | y n +2 − y ∗ | < k | y n +1 − y ∗ | < k | y n − y ∗ | and so on.We then see that if | r (cid:48) ( y ) | < y ∗ and if the starting point of therecursive sequence belongs to this same neighborhood, the distance to the fixed point reducesmore and more as n is increased and we’ll have lim n →∞ δ n = 0 meaning that lim n →∞ y n = y ∗ .In a more formal way, we state (without a complete and rigorous proof) the following theorem( fixed point convergence criteria ): If r ( y ) and r (cid:48) ( y ) are continuous on [ a, b ]2) if a ≤ y ≤ b → a ≤ r ( y ) ≤ b (meaning that r ( y ) is a contraction mapping )3) λ = max a ≤ y ≤ b | r (cid:48) ( y ) | < Then a) There exists a unique solution y ∗ ∈ [ a, b ] of the equation y = r ( y ).b) For any initial starting value y ∈ [ a, b ] the sequence will converge to the unique fixed point: lim n →∞ y n = y ∗ The convergence/divergence character of the fixed points can be interpreted graphically withthe so called “ cobweb (cid:48)(cid:48) construction.In the following Fig. 4 we have the recursion equation y n +1 = r ( y n ) plotted with y n +1 as afunction of y n . The fixed points are the intersections between r ( y n ) and the line y n +1 = y n .Here we have two fixed points labeled P and P . The cobweb construction shows that P is an9ttractive fixed (stable) point whilst P is a repulsive (unstable) fixed point. This is due to thefact that | r (cid:48) ( y P ) | < | r (cid:48) ( y P ) | > Fig. 4: Cobweb iteration of a sequence with an attractive ( P ) and a repulsive ( P ) fixed points. In the case presented in this article we are interested in the convergence of the sequence offunctions y n +1 = x y n . Here the x variable should be considered as a parameter of the recursion equation whose variablesare the terms y n and y n +1 . In practice we have an infinite number of sequences, one for eachvalue of x .The fixed points ( y ∗ ) of these sequences are those for which it is y n +1 = y n that is those satisfyingthe equation y = x y Using the fixed point convergence criteria we must find the interval of the values of the y (andof the x ) for which the first derivative of x y has modulus less than 1, that is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddy ( x y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < y = e ln x y = e y ln x and calculate the following derivative, with respect to y : ddy (cid:16) e y ln x (cid:17) Using the fact that for the fixed points it is y = x y we have ddy (cid:16) e y ln x (cid:17) = e y ln x · ln x = x y · ln x = y · ln x = ln x y = ln y For the convergence it must then be | ln y | <
1, that is − < ln y < → e < y < e The corresponding values for the x (since it is x = y y ) are then e − e and e e .Then, if we use the fixed point convergence criteria, we can say that the convergence is assuredfor e − e < x < e e producing stable fixed points in the interval1/ e < y < e. To gain a deepest understanding of what the previous result actually means, we now switch toanother route more rich of visual elements.In the case of the sequence y n +1 = x y n we see that the recursion equation is a family ofexponential curves (think of the “ x ” as a parameter) and the search of the possible fixedpoints and their stability is rather simplified, mostly because these functions are strictlymonotonic (apart the banal case with x = 1).In order to simplify the notation let’s rename the variables as follows: y n +1 = z, y n = y Then, we want to study the family of exponential functions z = x y (where the base x can be considered a parameter).With this notation a fixed point y ∗ is the solution of the system (cid:40) z = x y z = y leading to theequation y = x y . 11he character of the exponential is determined by the value of its base x : • x > • x = 1 : the exponential becomes the constant line z = 1 and the original infinite powertower function becomes y = 1 ... = 1; • x < z = x y with respect to the identityline z = y allow us to determine the possible existence of fixed points.With x > y ∗ (or twocoincident fixed points).III) The exponential curve intersects the line in two points: there are two distinct fixed points y ∗ and y ∗ . Fig. 5: : Possible relative positions of the exponential z = x y and the line z = y in the case x > With x < − ≤ ddy ( x y ) ≤ ddy ( x y ) < − ig. 6: The first derivative (slope of the tangent line) in the single fixed point in the case 0 < x < Now we’ll examine above 5 cases, analyze the characteristics of the fixed points and find whichvalues of the “ x ” produce them.If x > Fig. 7: : x > z = y So, let’s look for the tangency point T . In this point the exponential will have the same slopeof the line, that is z (cid:48) ( y ) = ddy ( x y ) = 1.Then z (cid:48) ( y ) = ddy ( x y ) = ddy (cid:16) e y ln x (cid:17) = e y ln x · ln x = x y · ln x = 1 → y T = log x (cid:18) x (cid:19) = − ln ln x ln x z T = x y T = x log x (1/ln x ) = 1ln x We have found the point T (cid:16) − ln ln x ln x , x (cid:17) in which it is z (cid:48) ( y ) = 1. But for the exponential curveto be tangent to the line z = y we must impose that T belong to that line, that is y T = z T → − ln ln x ln x = 1ln x → ln ln x = − → ln x = e − → x = e e ≈ . . With this value the exponential function becomes z = (cid:16) e e (cid:17) y and the point of tangency is T ( e, e ).Knowing how the base influences the graphic of a generic exponential curve we can also saythat:If x > e e there’s no intersection (and no fixed points for the recursive sequence).If x = e e there is a single intersection (and a single fixed point for the recursive sequence).If 1 < x < e e there are two intersections (and two fixed points for the recursive sequence).With the cobweb diagram we can see what evolution the sequence will follow in these cases:If x > e e (Fig. 8) there is no fixed point and, with any starting point, the sequence is boundto diverge to infinity.If x = e e (Fig. 9) the cobweb iterations converge to P ( e, e ) if the starting value is to theleft of P and diverge if the starting value is to the right. We can say that P is a “half-stable”saddle fixed point. Anyway, for the power tower sequence the starting value is y = x = e e that is located to the left of P ( e, e ). So the sequence converge to y ∗ = e . Fig. 8: Cobweb diagram in the case x > e /e Fig. 9: Cobweb diagram in the case x = e /e
14f 1 < x < e e there are two fixed points that are the solutions of the of equation y = x y . Let’scall them y ∗ and y ∗ (Fig. 10). Fig. 10: Cobweb diagram in the case 1 < x < e /e : the fixed point P is attractive and the fixed point P isrepulsive. The actual starting point ( x ) is in the basin of attraction of P The cobweb iterations show that y ∗ is attractive and y ∗ is repulsive. Furthermore the sequencewill converge to y ∗ for any starting point y < y ∗ and diverge to infinity for y > y ∗ . Anyway,for the power tower sequence the starting value is y = x and it’s located to the left of y ∗ . Infact, for the fixed point holds the relation y ∗ = x y ∗ → x = ( y ∗ ) / y ∗ and if we set x < y ∗ itmust be ( y ∗ ) / y ∗ < y ∗ . Taking the logarithms of both sides we have ln ( y ∗ ) / y ∗ < ln y ∗ that is1/ y ∗ ln ( y ∗ ) < ln y ∗ → y ∗ < → y ∗ > x < y ∗ if y ∗ >
1. But since z = x y is increasing and it’s z (0) = 1, the first intersectionof the exponential with the line z = y must have a value z >
1. This implies (since y = z ) that y >
1. So it is y ∗ > x < y ∗ . The sequence converges to y ∗ .To complete our analysis let’s see what happens with 0 < x <
1. In this case the exponentialcurve z = x y is decreasing and there can be only one single intersection point with the line z = y and a corresponding single fixed point. Anyway some interesting unexpected things are going tohappen when we start analyzing the stability of that fixed point and the eventual convergenceof the sequence to it.Two different cobweb iteration are presented for this case in the following figures, producingrather different outcomes. If the first derivative is | z (cid:48) ( y ∗ ) | < − ≤ z (cid:48) ( y ∗ ) <
0, Fig. 11)the iterations converge to y ∗ , oscillating between values alternatively greater and less than thatof the fixed point. We can say that the fixed point is attractive and that the sequence willeventually converge to it, whatever is the starting point.On the contrary, if the first derivative is | z (cid:48) ( y ∗ ) | > z (cid:48) ( y ∗ ) < −
1, Fig. 12) the iterations15re again oscillating but the sequence doesn’t converge to y ∗ . Instead it stabilizes towards aperiodic stable cycle, getting closer and closer to two alternate distinct fixed values.Let’s then see for what value of x we have z (cid:48) ( y ∗ ) > −
1. Since we have already found that z (cid:48) ( y ∗ ) = x y ln x we must solve the inequality x y ln x > − y = x y meaning x = y y . It willthen be y ln y y > − → ln y > − → y > e − → x = y y > e − e We can then say that the fixed point is attractive for e − e ≤ x < < x < e − e . Fig. 11: Cobweb iterations in the case 0 < x < − ≤ z (cid:48) ( y ) < < x < z (cid:48) ( y ) < − What can we say about the two values y and y involved in the 2-cycle?Since y is the next value in the sequence after y and y is the next value in the sequence after y we have y = x y and y = x y . Let’s take the power y of both sides of the second equationto get y y = x y y .Inserting x y = y we have y y = y y and y ln y = y ln y .Let’s now say that y is p times y , that is y = py and solve for y . py ln py = y ln y p (ln p + ln y ) = ln y → ( p −
1) ln y = − p ln ( p ) → ln y = p ln p − p → lny = ln (cid:16) p p − p (cid:17) We finally have y = p p − p and y = py = p p − p +1 = p − p . For instance, if we set p = 2 we have y = 1/4 and y = 1/2. These are the two values of thecycle that we’d get with x = y y = y y = (1/2) = (1/4) = 1/16 = 0 . extreme cycle in which y and y have the maximum separation. Setting p → ∞ we have y = lim p →∞ p p − p → ∞ − = 0 + y = lim p →∞ p − p = lim p →∞ e − p ln p → e − = 1 − and, since y = x y , 0 = x , x → Values of x Fixed point values Fixed point(s) Asymptotic behavior x > e e // No fixed points Divergence to + ∞ x = e e y = e < x < e e < y < e x = 1 y = 1 1 fixed point Instantaneous convergenceto the f.p. e − e < x < e < y < e x = e − e y = 1/ e < x < e − e < y < e < y < y and y Convergence to the 2-cycle x → + y → y → e ≈ . e e ≈ . /e ≈ . e − e = 1/ e e ≈ . It’s interesting to note that how the number e appears in above table in many possible powervariations.In conclusion, we can now say that the infinite power tower converges to the function definedby the expression y = x y (or x = y y ) for e − e ≤ x ≤ e e assuming values 1/ e ≤ y ≤ e .Taking into account the information collected we can show, in Fig. 13, the final plot of theinfinite power tower function. 17 ig. 13: Plot of y = f ( x ) = x x xx ... We have seen that the infinite power tower converges for e − e ≤ x ≤ e e , assuming values1/ e ≤ y ≤ e .But what happens outside the convergence interval?Let’s try some numerical experiment with some power towers with finite (but rather high) height.For x > e e the function f ( x ) blows out rapidly to + ∞ (Fig. 14). In fact we already know thatthere aren’t fixed points for the sequence y n +1 = x y n when we use x > e e .As we have already seen, for 0 < x < e − e the sequence start to oscillate between two boundedvalues, and some numerical simulation confirms that behavior (Fig. 15). The upper/lowerbranches of the plot correspond to an even/odd value for the height of the tower.We have already seen this oscillating behavior when exploring, through the cobweb diagrams,the recursive sequence y n +1 = x y n with x < e − e .18 ig. 14: Plot of y ( x ) = P owerT ower [ x, y ( x ) = P owerT ower [ x, y ( x ) = P owerT ower [ x, Let’s analyze further the origin of this feature.First we can try to calculate the limit of the finite power tower when x →
0. Let’s start with f : lim x → f ( x ) = lim x → x x = lim x → e x ln x we can now calculate the limit of the exponent (using the L’ H ˆopital’s rule)lim x → xlnx = lim x → lnx /x H = lim x → /x − /x = lim x → ( − x ) = 0 −
19o it is lim x → e x ln x = 1 − Then we are on the upper branch. What changes when we increase the tower height?lim x → f ( x ) = lim x → x x x = lim x → x ( x x ) and since lim x → x x = 1, as seen before, the last limit has the form 0 → x → f ( x ) = lim x → x x = 1 and lim x → f ( x ) = lim x → x x x = 0.We have a strong suspect (supported by the previous reasoning based on the cobweb diagrams)that these results may extend to towers with any height, with different values for n even and n odd, that is lim x → f n ( x ) = 1 and lim x → f n +1 ( x ) = 0 but we can’t prove this conjecture with simpletools and leave this problem to a later time.Having observed the oscillating character of the finite power tower sequence for 0 < x < e − e , weask ourselves if it’s possible to find the equations of the two distinct branches.Calling a and b the two values corresponding to some ¯ x it must be: y n ( x ) → a, y n +1 ( x ) → b, y n +2 ( x ) → a, y n +3 ( x ) → b, . . . This means that the sequence built with a double recursion should converge to its fixedpoints a and b .Let’s see what is the form of this double recursion: (cid:40) y n +1 = x y n y n +2 = x y n +1 ⇒ y n +2 = x x yn This sequence has stable fixed points if the derivative with respect to y of the right side hasmodulus less than 1, that is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddy (cid:16) x x y (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < x x y = e ln x xy = e x y ln x the derivative to calculate becomes: ddy (cid:16) e x y ln x (cid:17) after some passage we arrive at ddy (cid:16) e x y ln x (cid:17) = x x y + y ln x and it must be (cid:12)(cid:12)(cid:12) x x y + y ln x (cid:12)(cid:12)(cid:12) < RegionPlot command of
Mathematica we can visualize it (Fig. 16). Thedouble step sequence converges in the gray region and does not in the white one.20 ig. 16: RegionPlot of (cid:12)(cid:12) x x y + y ln x (cid:12)(cid:12) < In the gray region, where the double step sequence converge, it will converge to the sequencewhose fixed points are given by the transcendental equation y = x x y Now we can now put all the pieces together. 21 ig. 17: All the plots together: y = x y and y = x x y In Fig. 17 we can see the plots p and p defined by the equations y = x y and y = x x y respectively.These equations are also the equations defining the fixed points of the sequences s : y n +1 = x y n and s : y n +2 = x x yn . The gray region is where both sequences converge (“c” in the figure),while the white area (“d1” in the figure) is a region where there’s no convergence. The blue lineis produced by both equations (since the fixed points of a “single iteration” sequence are alsofixed points for the one with a “double iteration” step). At the left of the line x = e − e (“d2”in the figure) there’s no convergence for s and we have three branches. The upper and lowerbranches (in red) are produced only by the equation y = x x y .Since they lie in a region of convergence for this sequence their values can be also produced bythe infinite power tower function and we’ll have alternating values, one on the upper branch (foreven heights of the tower) and the other on the lower branch (odd values of the heights). Themiddle branch represent points produced by both equations. Anyway this branch is entirelylocated in the region “d1” where there is no convergence for both s and s . The infinite powertower won’t assume these values.Fig. 18 shows an enlargement of the region with the three branches.22 ig. 18: Detail of y = x y and y = x x y near the pitchfork bifurcation Now we can find an answer to our previously unanswered question: what is the limit of theinfinite power tower function when x → y = x x y , we havefor y → y → x → x x y = lim x → x x = lim x → e x ln x = 1for y → y → x → x x y = lim x → x = lim x → x = 0and the equation y = x x y is verified for both (cid:40) x → y → (cid:40) x → y → y n +2 = x x yn changes with different values of the x . Again, let’s use z = y n +2 and y = y n For e − e ≤ x ≤ e e our double step function z = x x y (solid line in the figures) has the same fixedpoints of z = x y (dashed line). Anyway, when x < e − e , two new intersections with the identity23ine appear Fig. (22). They correspond to the values of the stable 2-cycle. In the meantime,the fixed point y ∗ change from attractive to repulsive. In the theory of dynamical systems thetransition from one fixed point to three fixed points is called pitchfork bifurcation . Fig. 19: Graph of z = x x y for 1 < x < e /e Fig. 20: Graph of z = x x y for e − e < x < z = x x y for x = e − e . Fig. 22: Graph of z = x x y for 0 < x < e − e . Some history about the power tower
What is the origin of the power tower function? How come that someone had the idea ofcreating such a monster ? Actually its genesis can, somehow, be connected with the arithmeticaloperations based on Peano’s axioms :
1. zero is a number.2. if a is a number, the successor S ( a ) of a is a number.3. zero is not the successor of a number.4. two numbers of which the successors are equal are themselves equal.5. If a set K of numbers contains zero and also the successor of every number in K , then every number is in K (induction axiom). Peano’s axioms are the basis of the arithmetic of natural numbers, where the operations ofaddition, multiplication and exponentiation can be defined. Yet the only (unary) operationincluded in Peano’s axiom is the successor.However, we can build the other operations by iterating the one defined at the previous step.The operations defined in this way are called hyperoperations , and the grade 0 of this sequenceis the successor operation that, if iterated, can be used to define any natural number.So we can build the sequence of operations shown in the following table:
Name Definition hyper0 Successor S ( n ) = n + 1hyper1 Addition n + m = S m ( n )hyper2 Multiplication n · m = n + n + n + ... + n (cid:124) (cid:123)(cid:122) (cid:125) m hyper3 Exponentiation n m = n · n · n · ... · n (cid:124) (cid:123)(cid:122) (cid:125) m hyper4 Tetration m n = n n n...n (cid:124) (cid:123)(cid:122) (cid:125) m The sequence of hyperoperations can go on with the hyper5 (pentation), the hyper6 (hexation)and beyond.Naturally, the commonly used operations are the ones reaching hyper3 (exponentiation), but wecan see that the tetration is not just an exotic oddity but can be thought of as an extension ofthe process leading to the most usual arithmetical operations.The tetration with infinite height (infinite power tower) is often dealt together with the
LambertW function (called
ProductLog in Mathematica and
LambertW in Geogebra ).The Lambert W function y = W ( x ) is defined as the inverse function of x = y · e y (note thatthere’s no algebraic closed form expression for this function). http://mathworld.wolfram.com/PeanosAxioms.html x e x = 2. Its solution can be written as x = LambertW (2) and the numerical valuereturned is 0 . . · e . = 2).Taking advantage of the definition of the LambertW function, the fixedpoints of the infinite power tower can be expressed as y = x y = W ( − ln x ) − ln x In fact, starting from y = x y → y = e y ln x → ye − y ln x = 1 , multiply both sides by − ln x − y ln x · e − y ln x = − ln x set w = − y ln x ; z = − ln xwe w = z → w = W ( z ) that is, by definition, the LambertW function.Substitute back the w and z − y ln x = W ( − ln x ) → y = W ( − ln x ) − ln x that is the explicit form of theimplicit function defined by y = x y Johann Heinrich Lambert(1728-1777)
The definition of the Lambert W function originated by the article “
Observationes variae inmathesin puram” published in 1758 by the Swiss mathematician Johann Heinrich Lambert in which he dealt with the solution of the trinomial transcendental equation x m + px = q anddiscovered that, under certain conditions, the solution (a solution) could be expressed with thefollowing series: x = qp − q m p m +1 + m q m − p m +1 − m m − q m − p m +1 + m m −
12 4 m − q m − p m +1 − m m −
12 5 m −
23 5 m − q m − p m +1 To derive above series Lambert used a procedure that was later generalized by
Joseph-LouisLagrange in 1770 with what’s presently known as “ Lagrange inversion theorem ”.With Lagrange’s method, given a polynomial function y = f ( x ) = a x + a x + ... + a m x m it’s possible to find the series expansion of the inverse function x = g ( y ) = A y + A y + ... byapplying the following steps : • plug the first expression in the second x = A (cid:16) a x + a x + ... + a m x m (cid:17) + A (cid:16) a x + a x + ... + a m x m (cid:17) + A (cid:16) a x + a x + ... + a m x m (cid:17) + ... • equate the coefficients of the right and left sides having the same grade of the x . A a = 1 → A = a A a + A a = 0 → A = − a a A a + 2 A a a + A a = 0 → A = a − a a a ... ... Lambert J. H., (1758).
Observationes variae in mathesin puram , Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 128–168, 1758. Lagrange, Joseph-Louis (1770).
Nouvelle m´ethode pour r´esoudre les ´equations litt´erales par le moyen dess´eries , M´emoires de l’Acad´emie Royale des Sciences et Belles-Lettres de Berlin. 24: 251–326. If the polynomial contains a constant term a it’s possible to eliminate it with a change of variable y (cid:55)→ y − a http://mathworld.wolfram.com/SeriesReversion.html
26y finding the inverse function (or, better, an approximation of theinverse function around the point x = 0) it is also possible to find theapproximate value of a root ˜ x of a polynomial equation having the form f ( x ) = q since it is ˜ x = g ( f (˜ x )) = g ( q ).This procedure can be extended to generic (not polynomial) functions z = f ( w ) → w = g ( z ) using a more general form of the Lagrangeinversion theorem. Naturally there is the problem of convergence of theseries, problem that we won’t discuss here. Joseph-Louis Lagrange(1736-1813)
In a subsequent article, “
Observations analytiques” published in 1772, Lambert, examined thesimilar trinomial equation x = q + x m and wrote down the series that express not only a root ofthe equation, but also the powers of that root. In this article Lambert also mentions his meetingwith L. Euler in Berlin in 1764 and their discussions about the series connected with polynomialequations.Some years later, in 1779, Leonhard Euler published “
De serie Lambertina plurimisque eiusinsignibus proprietaribus ” in which, referring to the previous works by Lambert, he investigatedthe solutions of another trinomial equation, equivalent to the one studied by Lambert, havingthe form x α − x β = v ( α − β ) x α + β The equivalence can be verified by choosing the transformation of the parameters α = − m , β = − v ( α − β ) = q , obtaining1 x m − x = v ( α − β ) x m +1 → x − x m = q In this case the series useful to express the solution (or one ot its powers) is : x n = 1 + nv + 12 n ( n + α + β ) v + 16 n ( n + α + 2 β ) ( n + 2 α + β ) v + 124 n ( n + α + 3 β ) ( n + 2 α + 2 β ) ( n + 3 α + β ) v + ... Euler then makes a transformation of both expressions in the special cases α → β → α → β → ∧ n → x α − x β = v ( α − β ) x α + β ) it is x α − x β α − β = vx α + β → x α (cid:16) − x β − α (cid:17) α − β = vx α + β → x α (cid:16) x β − α − (cid:17) β − α = vx α + β and taking the limit β → α, β − α → ε, ε → ε → x α ( x ε − ε = vx α + α ⇒ x α ln x = vx α ⇒ ln x = vx α Lambert J. H., (1770).
Observations analytiques,
Nouveaux M´emoires de l’Acad´emie royale des sciences deBerlin, ann´ee 1770/1772 Euler L., (1779).
De serie Lambertina plurimisque eius insignibus proprietaribus, originally published in“Acta Academiae Scientarum Imperialis Petropolitinae” 1779, 1783, pp. 29-51 Euler’s series is not equivalent to Lambert’s because Euler’s series is centered around 1 and Lambert’s iscentered around 0. α →
1, ln x = vx .For the second expression it is x n − n = v + 12 ( n + α + β ) v + 16 ( n + α + 2 β ) ( n + 2 α + β ) v + 124 ( n + α + 3 β ) ( n + 2 α + 2 β ) ( n + 3 α + β ) v + ... and taking the limits for α → , β → , n → n → x n − n = ln x = v + v + 32 v + 83 v + ... Putting together the two expressions we can say that a special solutionof Euler’s trinomial equation can be written in two different ways:(1) the solution of ln x = vx and(2) the result of the series ln x = v + v + v + v + ... This means that the solution of the transcendental equation ln x = vx can be expressed by the series ln x = v + v + v + v + ... andif we set ln x = t (and x = e t ) we have t = ve t whose solution is t = v + v + v + v + ... The equation t = ve t can be rewritten as − te − t = − v and, using thedefinition of the Lambert W function as solution of we w = z ⇒ w = W ( z ) we have − t = W ( − v ) that is t = − W ( − v ) . So here we have theseries expansion Leonhard Euler(1707-1783)) − W ( − v ) = v + v + v + v + ... and W ( − v ) = − v − v − v − v + ... that is (setting − v = z ), W ( z ) = z − z + 32 z − z + ... representing the series expansion of the LambertW function.Even more closely related with the subject of this article is another work by Euler: “ Deformulis exponentialibus replicatis” , presented in 1777 (two years before the publication of“ De serie Lambertina ”), in which he investigated a problem posed by the French philosopherand mathematician Nicolas de Condorcet (known as Marquis de Condorcet), regarding theconvergence of the sequence r, r α , r r α , r r rα , ... The article’s opening is very interesting to point out Euler’s keen interest in such expressions.Its translation goes more or less like this: ”The famous Marquis de Condorcet recently shared with the academy deepspeculations regarding some rather unfamiliar analytic formulas, among which wecan, first of all, include the formulas called repeated exponentiations, where everypower goes into the power exponent following it. Yet, little has been achieved aboutthe nature of such expressions and despite the force of those investigations, led withincredible sagacity, no clear knowledge and perception has been reached. Hence itwill not be useless to explain here some special properties of such expressions.” Euler L., (1777).
De formulis exponentialibus replicatis, presented to the St. Petersburg Academy in 1777and published in “Acta Academiae Scientarum Imperialis Petropolitinae 1, 1778”. Also in Opera Omnia: Series1, Volume 15, pp. 268 – 297.
28n the article Euler proves that the sequence r, r α , r r α , r r rα , ... converges if e − e < r < e e .He also notes (p. 57) that the sequence β = r α , γ = r β = r r α , δ = r γ = r r rα , ... may producean alternate sequence of two values. In fact, choosing r = 1/16 and α = 1/2 we have β = r α = (1/16) = 1/4 , γ = r β = (1/16) = 1/2 , δ = r γ = (1/16) = 1/4and so on, with the results assuming the alternating values 1/2 and 1/4. Euler shows that, ingeneral, this happens when r Φ = Ψ and r Ψ = Φ leading to the identity Φ Φ = Ψ Ψ , since it is r Φ · Ψ = Ψ Ψ → (cid:16) r Ψ (cid:17) Φ = Ψ Ψ → Φ Φ = Ψ Ψ Now, this equation doesn’t necessarily imply that Φ = Ψ, because the function y = x x has aturning point for x = 1/ e and some y can be obtained with two different values of the x .To find the relation between the two values satisfying the equation Euler sets Ψ = p · Φ andfinds that it must be Φ = p p /(1 − p ) , Ψ = p − p ) and r = Φ (cid:16) = Ψ (cid:17) Finally Euler asks himself which is the condition for this two values to converge to a single value.This happens for p = 1 and it is Φ = lim p → p p /(1 − p ) = e − , Ψ = lim p → p − p ) = e − The corresponding value of r is r = Ψ = (1/ e ) e = e − e He then concludes that the relations r Φ = Ψ and r Ψ = Φ will always yield two different valuesif r < e − e .
10 Conclusions
We have considered the function based on a reiterated exponentiation y = x x x ... and haveinvestigated its properties, finding some counterintuitive fact. During our journey we had tocope with the unusual definition of this function, with its infinite sequence of exponents pilingup one over the others. To proceed forward and make some headway we had to use differentmathematical arguments, such as the concept of function and inverse function, limits andderivatives, exponentials and logarithms, sequences, fixed points of recursive sequences,cobweb diagrams and others. We also used experimental empirical tools like complexnumerical computations and graphical plots provided by mathematical software packages. Atthe end we can say that much of the properties characterizing the infinite power tower functionand its convergence (or not) to finite values have been explained.Anyway, what we are left with is a vague sense of awe and amazement in observing the mysteriousmetamorphosis of this function, from one leading to infinite results (as it was expected in the veryearly stages, before starting a more in-depth analysis), to one producing finite values and, lastly,to one undergoing some serious structural change (called bifurcation in the field of dynamicalsystems) and generating stable 2-cycles. We have used Euler’s method in Section 7. Corless R.M., Gonnet G. H., Hare D. E. G., Jeffrey D. J., Knuth, D. E., (1996).
On the LambertW function , Advances in Computational Mathematics 5, pp. 329-359.Knoebel R. A., (1981).
Exponentials Reiterated,
The American Mathematical Monthly Vol. 88,No. 4 (Apr., 1981), pp. 235-252Lynch P., (2017).
The Fractal Boundary of the Power Tower Function,
Proceedings ofRecreational Mathematics Colloquium V - G4G, pp. 127-138Lynch P., (2013).
The Power Tower Function, https://thatsmaths.files.wordpress.com/2013/01/powertowerlambert.pdf
Anderson J., (2004).
Iterated exponentials,
The American Mathematical Monthly Vol. 111, No.8 (Oct., 2004), pp. 668-679Glasscock D.,
Exponentiales replicatas (talk notes), http://mathserver.neu.edu/~dgglasscock/eulerexponent.pdf
Strogatz S., (1994).
Nonlinear dynamics and chaos , Westview Press. Chapter 10:“One-dimensional maps”Tetration (wikipedia): https://en.wikipedia.org/wiki/Tetration
Hyperoperation (wikipedia): https://en.wikipedia.org/wiki/Hyperoperation
Peano’s axioms (mathworld): http://mathworld.wolfram.com/PeanosAxioms.html
Series reversion (mathworld): http://mathworld.wolfram.com/SeriesReversion.html
Historical papers
Lambert J. H., (1758).
Observationes variae in mathesin puram , Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 1758, pp. 128–168Lagrange J. L., (1770).
Nouvelle m´ethode pour r´esoudre les ´equations litt´erales par le moyendes s´eries,
M´emoires de l’Acad´emie Royale des Sciences et Belles-Lettres de Berlin. 24, 1770,pp. 251–326Lambert J. H., (1770).
Observations analytiques,
Nouveaux M´emoires de l’Acad´emie royale dessciences de Berlin, ann´ee 1770/1772Euler L., (1779).
De serie Lambertina plurimisque eius insignibus proprietaribus , ActaAcademiae Scientarum Imperialis Petropolitinae, 1779, 1783, pp. 29-51Euler L., (1777).