Solving Infinite Power Tower Equation: Can we find a solution?

This video presents an infinite power tower equation, also known as a hyper-4 equation or exponentiation tower, and examines whether a solution can be found. This is also sometimes referred to as Tetration or SuperExponentiation or infinite left exponent. Repeated exponentiation in infinite power towers has a limited convergence range, so it is important to exercise caution when attempting to solve them. In general, Euler proved that the infinitely iterated exponential converges for exp(−e) ≤ x ≤ exp(1/e) that translates to approximately the interval from 0.066 to 1.44 for x. This interval includes the choice of x=sqrt(2).

[@STEMprof]

Thanks for watching & your feedback/comments (special thanks to @dariushsafari for the correction and review). For more examples please see Infinite Power Tower: ua-cam.com/video/JG1lg3aTig8/v-deo.html , Solving Infinite Tetration (Hyper4 operation) ua-cam.com/video/27VYkoUwA4w/v-deo.html ,
Power of Tetration ua-cam.com/video/AR2Rx_hbM1k/v-deo.html ,
and ua-cam.com/play/PLrwXF7N522y53xvU1h16_OH25ZCYZQpXy.html
and ua-cam.com/play/PLrwXF7N522y5BUuxzrned72krSgRWhTOh.html

[@erikev]

Could you convince me that this is sound math? How can you prove the limit converge? If it converges, how can you prove the solution unique?

[@STEMprof]

Thanks for watching and your comment. Please watch video starting at minute 6:50 when convergence condition is discussed. Generally speaking, repeated exponentiation in infinite power towers has a limited convergence range, so it is important to exercise caution when attempting to solve them. Euler proved that the infinitely iterated exponential in the form of hyper-4 operation x^x^x^...^x converges with a unique solution for exp(−e) ≤ x ≤ exp(1/e) that translates to approximately the interval from 0.066 to 1.44 for x. This interval includes the choice of x=sqrt(2) that is the solution of the super exponentiation power tower example discussed in this video. For more examples please watch: ua-cam.com/play/PLrwXF7N522y5BUuxzrned72krSgRWhTOh.html