Calculus 2, Lec 27A, Absolute vs Conditional Convergence, Taylor Polynomials and Series

Calculus 2, Lecture 27A.

(0:00) Career night announcement (and alternative extra credit).
(1:07) Lecture plans.
(1:27) An absolutely convergent alternating series.
(3:51) A non-absolutely (conditionally) convergent alternating series (the alternating harmonic series) and a remark about absolute convergence of power and Taylor series in the interior of the interval of convergence.
(6:11) Formula for Taylor polynomial of degree n centered at x = a for a "nice" function f, constructed in such a way as to match up the function and derivative values of f and its Taylor polynomial at x = a.
(10:42) Taylor series for f(x) = 1/(1+x^2) by recognizing it as the sum of a geometric series.
(13:49) Confirm answer with Mathematica.
(17:34) Program Mathematica to compute a general Taylor polynomial of degree n (should not have used "x" within "D"...that was the cause of the Mathematica errors).
(25:04) Find the Taylor series centered at a = 0 for the arctangent function via integration of 1/(1+x^2) and a representation of pi as an infinite sum.
(31:16) Differentiate the series for 1/(1+x^2) to get the Taylor series centered at a = 0 for -2x/((1+x^2)^2).