HUMBER Faculty of Liberal Arts & Sciences Q1. This question has two parts. (KB1) a) [4 marks] The exact value of " = 0.785398163397 ... can be estimated by the following series: (-1)n+1 = +1 2n - 1 How many terms of this infinite series must be taken so that the estimation of - is accurate within four decimal digits, i.e., leal $ 0.5 x 10-4. b) [5 marks] Find the third-degree Taylor polynomial P3 (x) for the f(x) = xex about Xo = 0. Approximate fo f(x) dx using fo Pa(x) dx, and find an upper bound for the error of this estimation. Note. The nth degree Taylor polynomial Pr (x) for the function f(x) at x, E (a, b) is defined by Pr (x ) = f ( kc) (x0 ) k! (x - x0)k K=0 The error of this estimation is bounded above by max (f(n+1) (x0) asxsb (n+1) 1 ( x - *:0 ) 12 + 1 1