Infinite Power Tower: Solving for Stability
We are given a fascinating recursive expression:
\[ x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 2 \]
This infinite power tower, also known as a tetration, asks:
What value of \(x\) causes this tower to converge exactly to 2?
Step 1: Understand the Definition
Let:
\[ y = x^{x^{x^{\cdot^{\cdot^\cdot}}}} \]
Given that \(y = 2\), and because the tower is infinitely recursive:
\[ y = x^y \]
Step 2: Solve the Equation
Substitute \(y = 2\) into \(y = x^y\):
\[ 2 = x^2 \]
Solving for \(x\):
\[ x = \sqrt{2} \]
Step 3: Validate Convergence
Infinite power towers don’t always converge. For convergence, it’s known that:
- The tower \(x^{x^{x^{\cdots}}}\) converges if and only if \(0 < x \leq e^{1/e} \approx 1.444\ldots\)
Check if \(\sqrt{2} \approx 1.4142\) is within that range — it is.
Thus, the tower converges and the result is valid.
Final Answer
\(\boxed{x = \sqrt{2}}\)