Question 2. Calculus (10 marks) 1. In the lectures, we skipped over showing that the function (1 + $)k was actually equal to the Taylor series m k ". 1 Z (n): N n=0 In principle, we needed to show that the remainder term vanished but this turns out to be very difcult. Instead, we can try something else: (a) The series in is some function of :E, as long as it converges. Let's call it 9(a) Differentiate the function g(;1:), show that the resulting series g'(:c) satises U+$MW=k, and give its radius of convergence. Hint: you can treat the lefthand side as multiplying polynomials termbyterm. You may make use of the identities: moan): (maids): (SH (3:. (4 marks) (b) Now, introduce a new function h(:r) = (1 + :E)_kg(:1:). Show that h'(:1:) = O. (2 marks) (c) Using the previous result, deduce that 9(33) 2 (1 + 21:)k, showing that indeed this function is equal to its Taylor series. Hint: What functions have a zero derivative for all values ofa: where they are dened? (2 marks) 2. Using the Taylor series expansion of ex, cos(:1:), and sin(:c), show that e" = cos(;1:) + isin(:1:) and deduce that ei7T + 1 = 0. (2 marks)