This Infinite Integral Tower Actually Converges? Sorta...

Take some function f(x) and compose it with itself over and over again, f(f(f(f(f.... What is the limit of that sequence? Certainly a converges for fixed points where f(x)=x, and indeed there is a whole theory about fixed point iteration that looks at the stability of fixed point. We've seen this in my previous video on cos(cos(cos(cos(... where the Banach Fixed Point Theorem guaranteed us a convergent sequence no matter where we started. In this video however, I've got a rather funky example, one where the f(x) is a particular integral and it has the golden ratio as a fixed point. However, it has all sorts of interesting dynamics beyond just the fixed points. My thanks to @DrBarker who first introduced me to framing this problem in terms of integrals. The real goal of my video is to elaborate on Dr. Barker's video and explain what happens in the domain -phi up to phi. His video: https://www.youtube.com/watch?v=xnFCncV-288 ►My Previous video on cos(cos(cos(cos(cos(cos....: https://www.youtube.com/watch?v=qHnXE_h5c2M 0:00 Intro 0:25 Defining as a sequence 2:41 Fixed Points 4:16 Where it diverges to infinity 6:10 -phi up to phi 8:30 Eventually Fixed Points 11:48 Summary 12:47 Math Merch!! COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw ►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 ►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography