SCHOOL OF MATHEMATICAL SCIENCES ENG1005 Engineering Mathematics Assignment 2: Infinite series Due date: Lab class for week 5 There are two questions in this assignment worth a total of 30 marks (6% of the final mark for this unit). The first question (16 marks) is a short answer question. You are expected to provide clear and concise answers. You will be marked mainly on the correctness of your answer though poorly presented or hard to read answers may loose marks. For the second question (14 marks), approximately half the marks will be for presentation. You will be marked not only on the correctness of your answer but also on the way you express yourself and in the way you layout your assignment. See the unit web site for examples of good and bad written assignments. For full marks you will need to show all of your working. Late submissions will be subject to late penalties (see the unit guide for full details). School of Mathematical Sciences Monash University Short answer question Use suitable tests to find which of the following infinite series converge and which diverge. Be sure to state which test you are applying. X n2 + 5 (a) 2n2 3 n=1 X 1 + ( 2)n (c) 2 + n n=1 X 3n (b) n3 n=1 4x4/16 X nn (d) n! n=1 Detailed answer question 2. You have two 3 litre vessels. The first contains oil while the second contains vinegar. You have two spoons, each can scoop out k litres of liquid, and you use the spoons to transfer k litres between each vessel. That is, you take a spoon from one vessel, a spoon from the other and then you transfer their contents over to the other vessel. You mix the vessels thoroughly so that each vessel contains a mixture of oil and vinegar. This process is then repeated any number of times. (a) Let Vn1 by the quantity of vinegar in the first vessel after n rounds of this game. Likewise, let Vn2 be the quantity of vinegar in the second vessel. Show that k 1 Vn+1 = Vn1 Vn1 + 3 k 2 Vn+1 = Vn2 Vn2 + 3 k 2 V 3 n k 1 V 3 n (b) Use the results of part (a) to show that 2 Vn+1 1 Vn+1 6/14 4/14 \u0012 2k =3 1 3 \u0013n+1 (c) Finally show that 4/14 1 Vn+1 3 = 2 2 = Vn+1 3 2 \u0012 \u0013n+1 ! 2k 1 1 3 \u0012 \u0013n+1 ! 2k 1+ 1 3 2