Infinite Power Tower (Tetration)

We want to solve the recursive equation x^{x^{x^⋱}} = 2. If the infinite tower converges to some value y, then by self-similarity it must satisfy y = x^y.

x^{x^{x^⋱}} = 2

Why the answer is x = √2

Let the entire infinite tower be y. Then the tower repeats itself after the first base:

Step 1

y = x^y

Step 2

We want y = 2, so 2 = x²

Step 3

Thus x = √2 ≈ 1.41421356

Fixed-point idea

The infinite tower is a fixed point of the map f(y)=x^y.

Target value

Because the tower must equal 2, we solve x²=2, giving x=√2.

Convergence range

For positive real bases, the standard infinite power tower converges when e^-e ≤ x ≤ e^1/e.

Interactive exploration

Base x 1.414214

Tower height n 10

Finite tower value

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Distance from 2

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Expected infinite limit

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Convergence interval check

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Move the slider to see how the finite tower behaves.

Successive finite towers

Starting from a₁ = x, define a_n+1 = x^a_n. If this sequence converges, its limit is the infinite power tower.

Height n	Expression	Value