Question 1: (3 points) Consider the series (i) What is the radius of convergence of this series? Write the letter i if the radius is infinite. (ii) Find the series expansion, centered at a: : 0 , for the derivative f! (cc) of f (at) . 4 What is the coefcient of a: in this series? (Do not use the symbol ! and give the exact value) (iii) What is the radius of convergence of the series for fr (3:) ? Write the letter i if the radius is infinite. (iv) Find the series expansion, centered at a: : 0 , for a primitive f f (:12) dm of f (:3) . What is the coefcient of 3:2 ? (Do not use the symbol ! and give the exact value) (v) What is the radius of convergence of the series for ff (:3) dm ? Write the letter i if the radius is infinite. Question 2: (3 points) Consider the series (i) What is the radius of convergence of this series? Write the letter i if the radius is infinite. (ii) Find the series expansion, centered at m : 0 , for the derivative f'(m) of f (m) . 7 What is the coefcient of m in the series for fits)? (iii) What is the radius of convergence of the series for f'(m) ? Write the letter i if the radius is infinite. (iv) Find the series expansion, centered at so 2 O , for the integral ff (m) dd: of f (m) . What is the coefcient of 3:5 ? (v) What is the radius of convergence of the series for f f (m) da: ? Write the letter i if the radius is infinite. Question 11: (1 point) The MacLaurin series expansions for the following two functions f and g begin thus: f (x) = 2 + 3x - 3x2 + 223 + ... g (x) = 3 - 2x + 2x2 - 2x3 +... Their product f (a) g () may also be represented by a MacLaurin series expansion f (x) g (x) = co + c1 x+ c2x2 + c323 +... What are its coefficients? 11 11 11 11