Infinite Power Tower  x^xxx··· = a

The math, step by step

1. The self-referential trick

Write the tower as y = x^xxx···. Look at the exponent sitting on top of the bottom x: it is another full copy of the same infinite tower. So the exponent equals y itself:

That one observation collapses an infinite expression into a single equation: x^y = y.

2. Solving for the base x

We are told the tower converges to some target a, i.e. y = a. Substitute and solve for x:

So for any target a you can find the base that produces it. For a = 2: x = 2^1/2 = √2 ≈ 1.41421.

3. Reading off the powers

Once you have x = a^1/a, the requested powers follow directly — and they have clean closed forms:

Notice x^a = a by construction, so when a = 2 the square lands exactly on the target: x² = 2. The cube is x³ = 2^3/2 = 2√2 ≈ 2.82843.

4. When does the tower actually converge? — full derivation ↓

Not every base works. The infinite tower settles to a finite value only when

which in terms of the target means 1/e ≤ a ≤ e (≈ 0.368 to 2.718). The upper limit e^1/e is the tallest base that still converges, and it produces the largest possible tower value a = e. Push x any higher and the iteration t ↦ x^t runs off to infinity — that's the red diverges badge.

▸ Where do the magic bounds e^−e and e^1/e come from? Click for Euler's derivation (parts a–e).

5. The √2 paradox (why 2 and not 4)

Here's the classic puzzle. The equation x^a = a with x = √2 is satisfied by two values: a = 2 and a = 4 (since √2²=2 and √2⁴=4). So does the tower equal 2 or 4?

The answer is 2. The tower is built by repeatedly applying t ↦ √2 ^t, and a value is only the real answer if that iteration is stable there — small nudges must shrink back, which requires the slope |x^t·ln x| < 1 at the fixed point. At a = 2 the slope is ln 2 ≈ 0.69 < 1 ✓ (stable, an attractor); at a = 4 the slope is 2·ln 2 ≈ 1.39 > 1 ✗ (unstable, a repeller). Starting from x and climbing, you always fall into the lower, stable root. That's why the chart above always converges to 2.

This is also why the curve below folds back on itself: each value of a below the peak is reached by two bases, but only the one in the green band gives a stable, convergent tower.

Step 4, in full — why only some bases work ↑ back to top