CIE 306 Code Call Me Maybe

1. Consider the following series and determine if they converge or diverge. In case of an alternating series specify if the convergence is absolute or conditional. (a) 1 X n=1 2n(n!) nn (b) 1 X n=1 (�1)n ln(n) n2 (c) 1 X n=1 (�1)n tan( 1 n) (d) 1 X n=1 npn n2 (e) 1 X n=1 pn(n2 + n + 1) n3 + 1 (f) 1 X n=1 n ln n 2n (g) 1 X n=2 (�1)n 1 n(ln n)2 2. Give examples of two series 1 X n=1 an and 1 X n=1 bn that both converge but such that the series 1 X n=1 anbn diverges. 3. For what values of x does the power series P (4x � 5)n n3=2 converge? Explain 4. (a) Find the McLaurin series for f(x) = ln(1 + x) and determine for what values of x the series converges. (b) Deduce lim x!0 ln(1 + x) x (c) Using the Mclaurin polynomial of degree 4, …nd an approximation for ln(2) 5. Find the Taylor series for f(x) = x3 � 2x2 + x + 1 with a = 1 6. (a) Using the series for ex; …nd the series for ex2: (b) Using the result in part a,express the integral R ex2dx as an in…nite series. 1