Exercise 6.6.10. Consider f(x) = 1/√1-x. (a) Generate the Taylor series for f centered at zero, and use Lagrange's Remainder Theorem to show the series converges to f on [0, 1/2]. (The case x < 1/2 is more straightforward while x = 1/2 requires some extra care.) What happens when we attempt this with x > 1/2? (b) Use Cauchy's Remainder Theorem proved in Exercise 6.6.9 to show the series representation for f holds on [0, 1).

find the taylor series for f(x)= 1/1-4x at x=2

1. Find the Taylor series for the functions sinh x and coshx for all x € R.

Given the below function and graph, determine the following. a. A Fourier series representation of the function. b. Write out the series explicitly with the first four terms after ao. 4, f(x) = { _x +4₁ -x 4, 3 2 1- −4 < x < 0 0

+ Find the first five non-zero terms of the Taylor series for f(x) = sin(2x) centered at 4π. + + + ...