The real power tower convergence interval
For a positive real base x, consider the sequence of finite, right-associated power towers:
The sequence converges to a finite positive real limit when:
Both endpoints are included. Convergence at the endpoints is especially slow, so a finite numerical simulation may not visibly settle even when the mathematical sequence converges.
Test a power tower base
Test whether a tower converges
Enter a positive base to see whether its sequence of finite power towers approaches a finite limit.
Tn+1 = xTn
The calculator checks successive values and compares the base with the theoretical convergence interval.
Where does an infinite real power tower converge?
The green-blue section represents the interval in which the standard infinite real power tower converges.
The bar is a linear illustration of the base interval from 0 to 5. It is not a graph of the tower's numerical value.
Example convergence classifications
| Base | Position | Expected behavior |
|---|---|---|
| 0.05 | Below lower boundary | Non-convergent |
| 0.1 | Small convergent base | Convergent |
| 0.5 | Inside convergence region | Convergent |
| 1 | Base one | Convergent |
| 1.3 | Inside convergence region | Convergent |
| 1.4 | Near upper boundary | Convergent |
| 1.5 | Above upper boundary | Non-convergent |
| 2 | Outside convergence region | Non-convergent |
Why does the convergence interval have these boundaries?
1. The fixed-point equation
If the finite tower sequence converges to a positive limit L, taking the limit in the recurrence gives:
Taking natural logarithms gives:
x = L1/L
2. Finding the largest possible base
Consider the function:
Its maximum for positive L occurs at L = e. The corresponding base is:
This gives the upper convergence boundary. At that point, the fixed point is L = e.
3. The lower boundary
The iteration is stable around a fixed point when the magnitude of the derivative of f(t) = xt is less than 1. Since:
The lower stability boundary occurs when L = 1/e, giving:
The endpoint cases are included in the convergence theorem, although their convergence is slower than in the interior.
What happens below the interval?
For positive bases below e-e, the standard sequence does not converge. Its terms typically settle into an alternating two-cycle rather than approaching one fixed value.
What happens above the interval?
Above e1/e, the fixed-point equation has no positive real solution. The successive finite towers grow without approaching a finite limit.
Frequently asked questions
What is the convergence interval for infinite tetration?
For the standard sequence of real power towers with a positive real base, the convergence interval is from e-e to e1/e, including both endpoints.
What is the approximate lower convergence boundary?
The lower boundary is e-e, approximately 0.065988. At this boundary, the tower converges to 1/e.
What is the approximate upper convergence boundary?
The upper boundary is e1/e, approximately 1.444668. At this boundary, the tower converges to e.
Does a power tower with base 2 converge?
No. The base 2 is above the upper convergence boundary. Its sequence of finite real power towers grows without approaching a finite limit.
Does a power tower with base 1 converge?
Yes. Every finite tower with base 1 equals 1, so the infinite tower converges immediately to 1.
Why does the calculator sometimes fail to detect convergence?
Convergence can be slow, particularly near either boundary. A finite number of iterations and a chosen numerical tolerance cannot always establish that the sequence has reached its limit.
Is an infinite power tower the same as a finite tetration?
No. A finite power tower has a specified height. Infinite tetration refers to the limit of a sequence of finite towers, when that limit exists.
Can negative bases be used?
This explorer is restricted to positive real bases. For negative bases, repeated real exponentiation is not generally defined as a real-valued sequence.