Tetration Base Convergence Analyzer

Why focus on the base of an infinite tower?

When you explore tetration, you likely begin with a finite tower such as a^aa and marvel at how quickly the numbers explode. AgentCalc already walks you through the finite case with its interactive tetration calculator. Yet many theoretical and practical questions revolve around the infinite limit: suppose the tower extends upward forever, does it settle on a number? The answer depends almost entirely on the base a. The convergence window 0 < a < e^1/e seems narrow, but even inside that range the behavior shifts from trivial convergence to delicate oscillations. Engineers who model iterated exponentials in feedback circuits, computer scientists studying super-recursive functions, and mathematicians analyzing complex dynamics all require a quick diagnostic. That is why this tool keeps its inputs grounded in the base, the seed, and conservative numerical safeguards. With a single run you receive a classification narrative, estimated limit, and an explanation of the underlying criteria, making it easy to cite your assumptions in papers or dashboards.

The calculator replicates a standard numerical experiment. You begin with a seed exponent x₀, apply the mapping xₙ₊₁ = a^xₙ, and repeat. If the sequence settles to a stable fixed point x*, the infinite tower converges. If the sequence jumps between a small set of values or explodes toward infinity, the tower diverges. Tracking the values requires careful floating-point handling because repeated exponentiation can overflow JavaScript’s Number type very quickly. When the magnitude of the exponent threatens to exceed 10³⁰⁸, the script halts early and flags the divergence. Likewise, extremely small values may shrink toward zero within machine precision, triggering convergence by underflow. The explicit tolerance field gives you control: smaller tolerances provide stricter convergence certification, though they may demand more iterations.

Analytically, the convergence question for real bases has a precise answer. The infinite tower converges for bases inside the interval (e^−e, e^1/e] and diverges otherwise. Inside the window, the limit solves the functional equation $x=a^x$ . Solving this transcendental relation typically requires taking logarithms on both sides and applying fixed-point iteration or Lambert-W functions. The tool implements fixed-point iteration because it mirrors the actual process of building the infinite tower. Engineers familiar with control theory will recognize the requirement that the derivative of the mapping stay below 1 in magnitude near the fixed point, ensuring contraction. That derivative is ln(a) · a^x, which leads directly to the analytic convergence bounds when you plug in the fixed point. By presenting both the numerical experiment and the theoretical thresholds in a single report, the analyzer shows how the pieces fit.

What the report includes

Each run generates a structured summary describing the base, the best available limit estimate, the numeric stability of the sequence, and the theoretical classification derived from the known bounds. If the numeric result agrees with theory, the description applauds the match; if not, it highlights the discrepancy so you can adjust your seed or tolerances. The summary also reports the iteration where convergence or divergence became apparent, along with the absolute difference between successive terms at that point. You can cross-reference these details with the multi-step table in the tooltips when you hover over the explanation, giving you deeper insight without overwhelming the main interface.

For educators and researchers, the analyzer doubles as a teaching prompt. Assign students different bases and seeds, then have them explain why some runs converge rapidly while others oscillate. Because the tool retains the last valid result even when you input nonsense, you can compare incorrect attempts with validated outputs. When the script encounters invalid data, it leaves the previous findings on screen and offers a friendly reminder about acceptable ranges. This is particularly important when demonstrating how sensitive the infinite tower is to seeds near zero or to bases above e^1/e, where a single iteration leaps out of range.

The following table compares different base values using the same seed and tolerance. It echoes the data near the top of the page but adds context on how many iterations are needed for convergence. Numbers are rounded to the fourth decimal place for readability.

Worked example: analyzing a base of 1.3

Suppose you suspect the base a = 1.3 produces a convergent infinite tower. You enter a seed x₀ = 1, keep the maximum iterations at 60, and set the tolerance to 10⁻⁶. The sequence begins at 1 and evolves to 1.3, 1.378, 1.427, and so on. After 43 iterations, the difference between successive terms drops below the tolerance, landing on x ≈ 1.3408. The summary reports “convergent,” citing both the numerical result and the theoretical bound because 1.3 is less than e^1/e. You then click through to the tetration calculator to build a finite tower with height six using the same base. Comparing the finite tower output with the infinite limit clarifies how quickly the sequence stabilizes: by the sixth level the value already matches the infinite limit to four decimal places. If you reduce the tolerance to 10⁻⁸, the calculator warns that more iterations are required but still converges, demonstrating how tolerance interacts with iteration count.

Now contrast that scenario with a = 1.8. Plugging the base into the analyzer produces rapid divergence. Within five iterations the sequence leaps beyond the largest representable double, forcing the script to stop and mark the tower as divergent. The report references the theoretical bound to show why: 1.8 exceeds e^1/e. Interestingly, if you lower the seed to 0.1 you may witness a few more iterations before overflow, but the long-term story remains the same. This comparison demonstrates why relying solely on finite tower experiments can mislead you; a base may behave nicely for small heights yet still have no infinite limit. The analyzer keeps your expectations grounded by embedding the theoretical context in its narrative.

Many readers wonder about bases below e^−e. For 0 < a < e^−e, the sequence collapses toward zero regardless of the seed, because the mapping exponentially squashes values. The tool highlights this by describing the convergence as “trivial collapse” when the limit falls below 10⁻⁶. That detail matters if you work with scaled towers in probability theory or fractal noise models: a base that seems innocuous actually kills the tower, leading to a degenerate distribution. Pairing the analyzer with the Bethe–Bloch energy loss calculator yields a fun physics exercise, too. The energy loss formula involves a logarithm nested within other functions; by substituting exponentials you can craft power towers that model hypothetical particle cascades. Checking the convergence ensures your scenario stays within physical limits.

Finally, the analyzer offers practical advice on reporting results. When the tool classifies the tower as convergent, it suggests quoting the limit to three significant digits unless you rerun with tighter tolerance and higher iteration caps. For divergent towers it lists the observed failure mode: overflow, oscillation, or alternating behavior. An oscillatory outcome often indicates a base slightly above 1 but with a seed that triggers even-odd alternation. The script flags this and explains how to average the two accumulation points if you are studying Cesàro means. These annotations turn the calculator into a micro-lesson that you can cite in documentation, grant proposals, or lecture slides.

Because the analyzer is entirely client-side, you can safely explore extreme seeds without sharing data. The validation logic blocks NaN or infinite inputs, enforces reasonable iteration counts, and catches tolerances that would produce meaningless conclusions. If a validation error occurs, the calculator preserves the last good summary so you never lose context mid-experiment. That reliability matches the user experience across AgentCalc’s most popular tools and makes this new entry a natural companion.