MATH 4010 HOMEWORK ASSIGNMENT V, 11/10/2015

MATH 4010 HOMEWORK ASSIGNMENT V, 11/10/2015 PROFESSOR CARLOS J MORENO Problem 20. (Textbook p.642 ) Find several terms in the power series expansion of the following quotients !(b) 1 cos(x) 1(c) 1 cos(x) - sin(x) Problem 21. (Textbook p.632 No. 8) Verify that the inverse hyperbolic sine function : sinh(x) = e x-e-x 2 has an inverse (i.e.x = sinh(y)or y = sinh-1 (x)) with a poer series expansion sinh-1 (x) = x + X8 n=1 (-1)n 1 · 3 · · ·(2n - 1) 2 · 4 · · ·(2n) · x 2n+1 2n + 1 (Justify your calculation and indicate for which valus of x you are proving the validity of the expansion, e.g. If the inteval of convergence is a finite interval, what can you say about the end points? Problem 22. Obtain the expansion T = x 2 + X8 n=2 (-1)n-1 1 · 3 · · ·(2n - 3) n! · x n 2 n for one root of the equation T 2 + 2T - x = 0, and show it converges so long as |x| < 1. Problem 23. Find the radius of convergence for the series X8 n=1 cnx n , where cn = 1 v n2 + 1 + 1 v n2 + 2 + · · · + 1 v n2 + n . 1