I would like to alert you to the amazing lectures in mathematical physics (perturbation theory) by Carl Bender (

http://physics.wustl.edu/cmb/).

On two occasions, he presents the fractals that depict the continued exponential (e^xe^xe^xe...) function, approximately speaking. To understand the context, you might need to rewind the lectures back to the beginning...

youtu.be/LMw0NZDM5B4?t=1h15m28s

youtu.be/YGDTW7mNUKM

... and indeed, it is worthwhile absorbing the entire course!

The first video in the series is found at youtube.com/watch?v=LYNOGk3ZjFM&list=PLOFVFbzrQ49TNlDOxxCAjC7kbnorAR1MU

You can watch the rest if you click on other lectures sorted by number on the right side of the YouTube page. (The videos come from

http://pirsa.org/index.php?p=speaker&name=Carl_Bender)

The rough outline of the course is to present the methods that solve the unsolvable things in physics. The perturbation theory is in focus. A hard problem (a function or equation) can be written as an infinite series of easy problems and solved in a few simple steps.

The infinite series can be convergent or divergent. Usually, the series is divergent, in which case a number of methods can be used to solve the sum: Euler, Borel, Zeta or Generic summation machines, continued functions, Pade summation, asymptotic series, Shanks' and Richardson transforms...

Various examples significantly lengthen the lectures, but you can sporadically encounter various remarkable statements such as 1 + 2 + 3 + 4 + ... = -1/12 in the 4th and 5th lectures, which explain how to sum a divergent series. Many recent YouTube videos and articles debate this topic in detail, so the topic attained considerable fame.

If you are only interested in the funny number series, check out these videos:

Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. by Mathologer

youtube.com/watch?v=jcKRGpMiVTw

Thoughts on the 1-1+1-1+... Series - A Gentle Discussion About Analytic Continuation by Mathoma

youtube.com/watch?v=UvX4JhGjRGk

Fourier Series From a Divergent Series by Mathoma

youtube.com/watch?v=EJLg5AAXjBQ

John Baez on the number 24

youtube.com/watch?v=vzjbRhYjELo

References:

https://www.scribd.com/doc/132045678/Bender-Orszag-Advanced-Mathematical-Methods-for-Scientists-and-Engineershttps://archive.org/stream/DivergentSeries/Hardy-DivergentSeries* NEW *

On the topic of a similar fractal type, Paul Bourke discusses the tetration fractals here...

http://paulbourke.net/fractals/tetration/"Tetration is repeated exponentiation."

Announcement about the new Mandelbrot deep zooming technique that uses the perturbation theory to approximate the calculations, by Christian Kleinhuis in chaosTube - news #14 - Compo2016 Announcement - Compo 2015 Review - Fractal Software

youtube.com/watch?v=6U0zi4Qe42M