The General Infinite Exponential and Its Limit

The iterated exponential x^x^x...^x was investigated here and here. Whenever x is in [(1/e)e,e(1/e)], the limit is given by Lambert's W function:

h(x) = -W(-log(x))/log(x) (1)

Can we say anything about the general infinite exponential a(1)^a(2)^a(3)^...^a(n)? Barrow in the Tetration References shows that this infinite exponential converges, if:

  1. Em in N: An >m, a(n) in [(1/e)e,e(1/e)].
  2. a(n) converges.

We will show the following theorem:

Theorem:

Whenever the general infinite exponential a(1)^a(2)^... converges, Em in N: limn->∞a(1)^a(2)^...^a(n) = a(1)^a(2)^...^a(m)^h(L), where L=limn->∞a(n).

Proof:

limn->∞a(n)=L, =>
A > 0, Em: L- < a(n) < L+, An >m. Depending on whether n-m is even or odd, we must have:
An >m, a(1)^a(2)^a(3)^...^a(m)^(L-/+)^(L-/+)^... (n-m times) < a(1)^a(2)^a(3)^...^a(n) < a(1)^a(2)^a(3)^...^a(m)^(L+/-)^(L+/-)^... (n-m times).

Now, the iterated sub-exponential (L+/-)^(L+/-)^...^(L+/-) converges to h(L+/-), by (1), hence again depending on whether n-m is even or odd,

A,1 > 0, Em,m1 in N: a(1)^a(2)^...^a(m)^{h(L-/+)-/+1} < a(1)^a(2)^a(3)^...^a(n) < a(1)^a(2)^...^a(m)^{h(L+/-)+/-1}, An >m+m1.

h(x) is continuous and monotone increasing in [(1/e)e,e(1/e)], so by choosing =|L-/+-L|, =>

A,1,2 >0, Em,m1 in N: a(1)^a(2)^...^a(m)^{h(L)-/+2-/+1} < a(1)^a(2)^a(3)^...^a(n) < a(1)^a(2)^...^a(m)^{h(L)+/-2+/-1}, =>
A3 >0, Em,m1 in N: a(1)^a(2)^...^a(m)^{h(L)-/+3} < a(1)^a(2)^a(3)^...^a(n) < a(1)^a(2)^...^a(m)^{h(L)+/-3}.

The function a(1)^a(2)^...^a(m)^x is continuous in [(1/e)e,e(1/e)], therefore by choosing 3=|h(L)-/+3-h(L)|, =>

A4 >0, Em,m1 in N: a(1)^a(2)^...^a(m)^h(L)-/+4 < a(1)^a(2)^a(3)^...^a(n) < a(1)^a(2)^...^a(m)^h(L)+/-4, hence,

A4 >0, Em,m1 in N: |a(1)^a(2)^a(3)^...^a(n)-a(1)^a(2)^...^a(m)^h(L)|<4, An >m+m1, and we are done.

Let's see some examples with Maple:

> b:=proc(L) #Calculate the general exponential
> option remember;
> local n;
> n:=nops(L);
> if n=1 then L[1];
> else
> L[1]^b(subsop(1=NULL,L));
> fi;
> end:

> tower:=proc(x,n) #Power tower of height n
> option remember;
> if n=1 then x;
> else
> x^tower(x,n-1);
> fi;
> end:

We first choose a sequence a(n), the power tower 1.25^1.25^...^1.25.

We first check that the limit indeed falls in [(1/e)e,e(1/e)].

> W:=LambertW;
> h:=-W(-log(x))/log(x);

> h(1.25);
1.352203514

OK. Falls in interval of convergence.

> findm:=proc(L,l,eps)
>#find m as in proof
> local m;
> m:=1;
> while evalf(abs(L[m]-l))>=eps and m < nops(L) do
> m:=m+1;
> od;
> m;
> end:

> L:=[seq(tower(1.25,n),n=1..30)]: #list of values for general exponential.

> m:=findm(L,h(L[1]),0.0000001);
m := 13

> Q:=L[1..m]:Q:=[op(Q),h(L[1])]:
> b(Q); #estimated limit from theorem
1.419809666

> b(L); #actual value of iterated exponential
1.419824956

A second example, with an arbitrary iterated exponential:

First set the limit M to be the midpoint in [(1/e)e,e(1/e)]

> M:=evalf((exp(-exp(1))+exp(exp(-1)))/2);
M := .7553279484

> a:=n->M+1/n^3; #sequence which converges to M.
> L:=[seq(evalf(a(n)),n=1..20)]:
> m:=findm(L,M,0.001);
m := 11

> Q:=L[1..m]:
> Q:=[op(Q),h(M)]:
> b(Q); #estimated limit from theorem
1.659143069

> b(L); #actual value
1.659143069

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