Question 3. (Show Working) 10 points

The Maclaurin series for a function is a special name for its Taylor series with centre 0.Consider the function f : [-1, ∞) →→→ R defined by f(x)=\sqrt{1+x} (a) Write down Taylor's theorem with Lagrange remainder for the specific case of fcentred at 0, for an arbitrary degree n. (b) Use the n = 2 version of Taylor's theorem to find a bound on the error if the degree 2 Taylor polynomial centred at 0 for f is used to approximate √1.05.

(c) Let x € (0, 1). Use Taylor's theorem from part (a) to show that the sum of the Maclaurin series of f evaluated at x equals f(x).

(d) Can you do the same as part (c) for x € (−1,0)? Why or why not?

(e) Write down the Maclaurin series for ƒ and show that it has radius of convergence 1.

(f) Write S(x) for the sum of the Maclaurin series for f evaluated at x € (−1,1). Show that, for x € (−1, 1), the series satisfies the differential equation 2(1+x) S^{\prime}(x)=S(x)

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6