35. Find the Taylor series expansions for the following functions, and determine when they converge: (a) Pr

35. Find the Taylor series expansions for the following functions, and determine when they converge:

(a) PðrÞ ¼ D r with r0 ¼ 0:05.

(b) f ðxÞ ¼ sin x with x0 ¼ 0. (Hint: Use (9.16).)

(c) gðxÞ ¼ cos x with x0 ¼ 0.

(d) Confirm using parts

(b) and (c), that in terms of the resulting Taylor series, eix ¼ cos x þ i sin x;

which is Euler’s formula from (2.5) in chapter 2.