Infinite Power Tower: Solving for Stability
We are given a fascinating recursive expression:
This infinite power tower, also known as a tetration, asks:
What value of causes this tower to converge exactly to 2?
Step 1: Understand the Definition
Let:
Given that , and because the tower is infinitely recursive:
Step 2: Solve the Equation
Substitute into :
Solving for :
Step 3: Validate Convergence
Infinite power towers don’t always converge. For convergence, it’s known that:
- The tower converges if and only if
Check if is within that range — it is.
Thus, the tower converges and the result is valid.
Final Answer