Question

a) Find the radius and interval of convergent of power series on(x+2) . 31+1 [4 marks] b) Find the Maclaurin series for the following function: i. sin x [3 marks 11, cos x [3 marks] 2

Please write in clear solution. Thanks.

Answer 1

Given sigma^infinity_n = 0 n (x + 2)^n/3^n + 1 Let's do the ratio test for absolute convergence Let a_n = n(x + 2)^n/3^n + 1 then Vertical-bar a_n + 1/a_n Vertical-bar = Vertical-bar (n + 1) (x + 2)^(n + 1)/3^n + 1 + 1 middot 3^n/n(x + 2)^n Vertical-bar = (n + 1)/9^n Vertical-bar x + 2 Vertical-bar as n rightarrow infinity rightdoublearrow Vertical-bar x + 2 Vertical-bar /9 from this we see that the power series will converges absolutely y Vertical-bar x + 2 Vertical-bar /9 < 1 from there we can see that the Radius of convergence is R = 9 and the interval of convergence is - 9 < x + 2 < 9 - 11 < x < 7 sin (x) Let f(x) = sin(x) To find Maclaurin series we have to find the coefficients (d^k/d^x^k sin (x)) Vertical-bar _x = 0 for k = 0, 1, 2, 3, 4 \r\n f(0) = sin (0) = 0 f^1 (0) = cos (0) = 1 f^11 (0) = -sin (0) = 0 f^111 (0) = -cos (0) = -1 f^1111 (0) = sin (0) = 0