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Do infinite power towers actually converge? In this video, we compare two fascinating expressions: i^(i^(i^(…))) and (−i)^(−i^(−i^(…))) At first glance, both towers look chaotic. But using complex analysis, Euler’s formula, and the Lambert W function, we uncover their hidden structure and determine whether they converge — and if so, where. You’ll see: • How infinite exponentiation becomes a fixed-point problem • Why logarithms turn iteration into geometry • How the Lambert W function solves equations of the form u · e^u = k • Why both towers converge to complex conjugates • A numerical visual verification showing convergence step by step This video combines rigorous mathematics with clean visual intuition, making advanced ideas in complex analysis both accessible and visually compelling. 🧠 Topics Covered Infinite power towers Complex exponentiation Fixed points and convergence Euler’s formula Lambert W function Complex conjugates Iterative maps in the complex plane 🔍 Why This Is Interesting Infinite exponentiation behaves very differently in the complex plane. Even a small change — replacing i with −i — leads to a deep and surprising result. 🏷️ Tags / Keywords infinite power tower, complex exponentiation, i power tower, minus i power tower, lambert w function, complex analysis, convergence, fixed point iteration, euler formula, complex conjugates, math visualization, manim animation, advanced mathematics