   Taylor Series Computations, Graphs of Partial Sums, Ratio Test by
Video Lecture 28 of 35
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Date Added: July 29, 2017 Lights Out New Window

### Lecture Description

(0:00) Quiz ready. (0:22) Purely algebraic proof of the Fundamental Theorem of Algebra (involves Field Theory, Galois Theory, Group Theory, and the Sylow Theorems). (2:32) Plan for the day. (3:11) Taylor series for an analytic function centered at a point z0 (at which it is analytic). (6:58) Find the Taylor series for the sine function centered at z0 = 0. (12:07) Show graphs of partial sums on the real line converging to the graph of the sin(x). (14:40) Write down the Taylor series for the cosine function centered at z0 = 0 and graph the partial sums on the real line. (16:36) Write down the Taylor series for the exponential function e^z centered at z0 = 0 and graph the partial sums on the real line. (18:47) Taylor series for 1/(1 - z) centered at z0 = 0 (geometric series), for z within 1 unit of 0. (21:18) Use the Ratio Test to prove the convergence of the Taylor series for e^z for all z (though this doesn't prove it converges to e^z) and graph the partial sums on the real line. (29:41) Taylor series for 1/(1+z^2). (32:21) Graph of partial sums on the real line. Why is the interval of convergence just the open interval from -1 to 1? The function has a pole at z = i and z = -i. (38:28) Integrate the Taylor series for 1/(1+z^2) term-by-term to get the Taylor series for the principal value of the arctangent function. (42:29) Approximations of pi with the Taylor series for arctangent (including Machin's formula). (46:56) Differentiate the series for 1/(1+z^2) term-by-term to get the Taylor series for -2z/((1+z^2)^2).

### Course Description

Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.

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