Intro Real Analysis, Lec 29: The Most Beautiful Equation in the World, Taylor Series Calculations

Introduction to Real Analysis, Lecture 29.

(0:00) Start
(0:27) What is the most beautiful equation in the universe? It is e^(i*pi)+1=0. It has the five most important numbers, the three most important operations, and the most important equivalence relation in one simple statement.
(2:31) Why is it true? Because of Euler's identity, which can be derived using the Taylor series for e^x, sin(x), and cos(x). It is assumed that the series can be combined by rearrangement (it can be done because they are absolutely convergent series).
(9:33) Visualize Euler's identity in the complex plane with Mathematica (and in terms of the Taylor expansion of e^(x)).
(17:20) We will continue focusing on calculations today. The Taylor series for e^(x) as a summation (with sigma notation). Confirm that it converges for all x with the Ratio Test (so the interval of convergence is from -infinity to infinity and the radius of convergence is infinity).
(22:53) Taylor series for cosine and sine in summation form.
(25:39) Find the Taylor series for x^2*sin(x^3) by substitution and multiplication. Check by using the Mathematica function Series.
(30:01) Know the Taylor series for 1/(1-x) (based on the geometric series formula) and be able to find Taylor series for related functions by substitution (such as for 1/(1+x) and 1/(1+x^2)). The interval of convergence in each of these cases is the open interval (-1,1).
(33:41) Integrate 1/(1+x) term by term to get the series for ln(1+x). This happens to converge on the half open half closed interval (-1,1] (when x = 1 the series converges conditionally (non-absolutely) to ln(2), but the convergence is slow).
(38:54) The convergence will be uniform on any proper closed subinterval.
(39:44) Integrating the Taylor series for 1/(1+x^2) term by term gives the series for the arctangent function arctan(x) = tan^(-1)(x). This also allows us to get an infinite series expansion for pi (with a slowly converging alternating series).
(42:35) Differentiate the Taylor series for 1/(1-x) = (1-x)^(-1) to find the Taylor series for 1/(1-x)^2 = (1/(1-x))^2. This can also be confirmed by multiplication of Taylor series.
(46:45) Use algebraic tricks to find the Taylor series for 5/(7+3x) centered at x = 6. Check it with the Mathematica function Series. The interval of convergence is for those values of x within 25/3 of the number 6.
(52:11) Comments about two homework problems 1) related to pointwise convergence of the sum of two pointwise converging sequences of functions and 2) related to showing that absolute convergence implies convergence.

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@17:00 would the graph eventually display a chaotic behavior (bifurcation, maybe) as theta increases?