(9/1/08)Math 10B. Lecture Examples.Section 9.1. Sequences†Example 1 A piece of meat at 30◦C is put in a freezer at time n = 0. The temperature of thefreezer is 0◦C, and the temperature of the meat n hours later is T =30n + 1(Figure 1).Does the sequence30n + 1∞0as n → ∞ converge? If so, what is its limit?n10 20T◦C102030FIGURE 1Answer: limn→∞30n + 1= 0 converges and its limit is 0. (The temperature of the meat approaches the temperatureof the freezer as n → ∞.)Example 2 Figure 2 shows t h e graph of the population P = 1000(2n/6) on day n of a colony ofbacteria that consists of 1000 bacteria at n = 0 (a) How long does it take for thepopulation to double? (b) Does the sequencen1000(2n/6)o∞n=0converge?n10 20P5,00010,000FIGURE 2Answer: (a) The population doubles every 6 days. (b) •1000(2n/6)∞n=0diverges.†Lecture notes to accompany Section 9.1 of Calculus by Hughes-Hallett et al.1Math 10B. Lecture Examples. (9/1/08) Section 9.1, p. 2Example 3 Figure 3 shows the graph of the t h e number of days y = dnin February of yearn ≥ 2000: dnis 29 for leap years n when n/4 is an integer and is 28 other years. Whathappens to the sequence {dn}∞n=2000as n → ∞?ny302728292000200420082012FIGURE 3Answer: {dn}∞n=2000diverges.Example 4 Does the sequencene1/√no∞n=1converge or diverge? If it converges, give its limit.Answer: The sequence converges and its limit is 1. (The table below shows that the limit is approached relativelyslowly.)n1 10 100 1000 10,000 100,000e1/√n.=2.7183 1.3719 1.1052 1.0321 1.0101 1.0010Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 10.1: Examples 1–5‡The chapter and section numbers on S henk’s web site refer to his calculus m anuscript an d not to the chapters and sectionsof the textbook for the