1. Apply Taylor's Theorem directly to find the power series for the function and determine the radius of convergence: (a) f(x) = e-* at x = 1 f" find pattern (c) f(x ) = est at x = 0 ( b ) f ( 20 ) = 1 at x = 0 1 + x ifn "(1) = patted f(ac) = Vx at x = 4 1 1 2. Use the power series for 1 - x to find the power series for the function f (x) = 1 + 24 3. Use the power series for ed to find the power series for the function f(x) = 3x2ex's. (Hint: There are at least two different ways of doing this. See if you can find both of them-start by finding the power series for exs.) 4. Consider the function f(x) = sin x. (a) Apply Taylor's Theorem directly to find the power series centered at x = 0. (b) Differentiate your power series from the previous part to find the Taylor series for g(x) = cos x. (c) (Optional-But Really Cool) Use the Taylor series for ed, sin x, and cosx, to achieve the famous Euler formula in complex analysis: ex = cosx + isin x