Sample1
- 1. Merit Worksheet 10. Math 231 AD1. Spring 2016. 3/2/2016. 11.1: Sequences 1. What do you conclude from the discussion we just had? 2. Now think about the following scenario: You have a lamp in your room. At 10pm you decide to go to bed so you turn the lamp off. At 11pm you realize you forgot to do your Math 231 homework for next morning and so you turn the lamp on again. At 11.30pm you finish your homework and so you turn the lamp back off. At 11.45pm you realize you forgot one part of the homework and so you turn the lamp back on. At 11.52pm and 30 seconds you finish that part and go back to bed. But after 3 minutes and 45 seconds you again have to get up. This pattern continues. Regardless of how many times you go back to bed and turn the light off you have to get back up because you remember something was missing and for some mysterious reasons the time intervals keep getting halved. Question: At 12 pm is the light on or off? What does your answer tell you? 3. All the following triangles here are equilateral triangles: a) What is the relation between the length of the line AC and the combined length of AB +BC? b) What is the relation between the length of the line AC and the combined length of AD + DE + EF + FC? c) What is the relation between the length of the line AC and the combined length of AG + GH + HI + IE + EJ + JK + KL + LC? d) Do you see a pattern? Do you have any guess for what will happen when we make the line more jagged? e) What will happen if we continue this indefinitely? What weird conclusion does that suggests?
- 2. Now that we have seen why we should study sequences let’s start: 4. Which one of the following sequences is convergent? Try to justify your answer in your group a) an = 1 n b) bn = (−1)n c) cn = (−1)n n d) dn = n n2 + 1 e) en = n2 2n f) fn = n! 3n 5. Order the following sequences in order of growth towards infinity: 5n , 36, 3n! + 2, 4n + 5, log(n), nn , n3 − 8, log(n)n 6. Most awesome thing about sequences: Giving meaning to infinite sums!!! We have the following summation: ∞ n=1 an = a1 + a2 + a3 + a4 + ... We will build a sequence out of it: 1. Let s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3 2. What do you think sn is? 3. What does the convergence of the sequence sn tell you? 7. Let’s use the previous question: a) Let a1 = 1 b) Let a2 = 1 − 1 = 0 c) Let a3 = 1 − 1 + 1 = 1 d) Based on this pattern what is an? e) Does an converges? f) What does this tell you about the following sum? ∞ n=0 (−1)n = 1 + (−1) + 1 + (−1) + ... g) In light of your answer what can you say about the following intuitive equality: 0 = 1 + (−1) + 1 + (−1) + ... = ∞ n=0 (−1)n