Calculate Infinite Tetration
An infinite power tower repeatedly raises a number to the power of itself: xxx.... Enter a positive base to investigate whether the tower converges.
Understanding Infinite Tetration
Tetration is the repeated exponentiation of a number. While ordinary exponentiation calculates expressions such as xy, tetration creates a tower of exponents.
x ↑↑ 2 = xx
x ↑↑ 3 = x(xx)
x ↑↑ ∞ = xxx...
When Does an Infinite Power Tower Converge?
For a positive real base x, the standard real convergence interval for the infinite power tower is:
The approximate numerical limits are:
- Lower bound: 0.065988035845
- Upper bound: 1.44466786101
Bases inside this interval have a finite infinite-tetration limit. Bases outside it do not produce a convergent finite real power tower under the standard repeated-tower definition.
The Fixed-Point Equation
If the infinite tower converges to a value T, removing the first level of the tower does not change its limit. Therefore:
This equation is used to describe the limit. The calculator approximates it by repeatedly evaluating finite towers until consecutive values are sufficiently close.
Why Are the Boundary Values Special?
At the upper boundary x = e1/e, the limit is e. At the lower boundary x = e−e, the corresponding fixed-point value is 1/e. These are critical cases where ordinary numerical iteration can converge very slowly or behave marginally.
Numerical Precision
This calculator uses PHP floating-point arithmetic. It provides numerical approximations rather than arbitrary-precision exact values. Near the convergence boundaries, a large number of iterations may be needed to reach a small tolerance. A result that reaches the iteration limit without satisfying the tolerance should be treated as an approximation, not as a fully converged numerical result.
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