Question

1. Given the series -1)" n! , 2n+1 (2n1) (i) Find the radius of convergence of the series. (ii) Find also the largest open interval on which the series converges. 2. (a) Find the Taylor series, in summation form, of f(x) = 1+1 (b) (i) (ii) Find the radius of convergence of the series. Find also the largest open interval on which the series converges. 3. (a) Find two series solutions of the differential equation +9=0, -oo < x < oo. dx2 (b) Give a sketch, on a graph, of the representation of the series solutions of the differential equation NOTE: You can show any value of n, which is the degree of the approximated polynomial.

Answer 1

"Sigma (-1)^n n!/(2n + 1)! middot x^2n + 1 a_n = (-1)^n middot n!/(2n + 1)! middot x^2n + 1 lim Sup