M AT H 1 2 0 Differential and Integral Calculus Winter 2016 - Problem Set #20 Due Thursday, March 17th. Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your nal write-up is clean and clear and effectively communicates your reasoning to the grader. 1) Consider the series 3n x n (1)n+1 n4n where x R. Find all values of x such that the series converges. n =1 2) For each of the following series if it converges absolutely, if it converges conditionally, or if it diverges. (a) 2k k! k =1 (b) (1)n+1 3n 1 n =1 (c) m2 m! (2m)! m =1 (d) 3i + i i! + 2 i =0 (e) sin( j ) , j2 j =1 R 2 (f) (1)l l 2 4 + l2 l =0 (g) (1) p+1 ln( p) p =2 3) Suppose that ( an ) 1 is a sequence such that an 0 for all n 1. n= n =1 (a) Show that if the series n =1 an converges, then the series ln(1 + an ) converges. n =1 (b) What about the converse? Is it true that if n =1 ln(1 + an ) converges, then an converges? If it is true, give a proof. If it is false, give a counterexample