Calculus 2, Lec 29A, Taylor Series for ln(x) about x = 1, Error Bounds, Series for sin(x)

Calculus 2, Lecture 29A.

(0:00) Lecture plans.
(0:20) Example 1: Taylor series for f(x) = ln(x) centered at a = 1 by using the derivatives' values at 1.
(7:07) Example 2: Find the same Taylor series by integrating 1/x after representing 1/x in a Taylor series centered at a = 1 and the find the interval of convergence.
(14:13) Graphically estimate the error in using the 3rd degree Taylor polynomial for ln(x) centered at a = 1 over the interval from 0.1 to 1.9.
(21:36) Lagrange error bound statement and application over the interval [0.1,1.9] to f(x) = ln(x) and its Taylor series of degree 3 centered at a = 3 (actual error is much smaller).
(29:55) Example 3: apply the Lagrange error bound to the sine function to conclude that the Taylor series for the sine function actually equals it for all x.