We are given a fascinating recursive expression:

xxx=2 x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 2

This infinite power tower, also known as a tetration, asks:

What value of xx causes this tower to converge exactly to 2?

Step 1: Understand the Definition

Let:

y=xxx y = x^{x^{x^{\cdot^{\cdot^\cdot}}}}

Given that y=2y = 2, and because the tower is infinitely recursive:

y=xy y = x^y

Step 2: Solve the Equation

Substitute y=2y = 2 into y=xyy = x^y:

2=x2 2 = x^2

Solving for xx:

x=2 x = \sqrt{2}

Step 3: Validate Convergence

Infinite power towers don’t always converge. For convergence, it’s known that:

  • The tower xxxx^{x^{x^{\cdots}}} converges if and only if 0<xe1/e1.4440 < x \leq e^{1/e} \approx 1.444\ldots

Check if 21.4142\sqrt{2} \approx 1.4142 is within that range — it is.

Thus, the tower converges and the result is valid.

Final Answer

x=2\boxed{x = \sqrt{2}}

Reference