Exercise 4.14.1.1. Consider the sequence an 2n and the claim that lim 1 = L = 0. n-too 2n a.) For the limit above, find minimum N values for each of the following c. That is, for each E, find the smallest value of N such that all Jan - L| N. . E = 0.1 . E = 0.01 . E = 0.001 b.) Find a formula for N in terms of c. Plug in e = 0.01 and confirm your answer above. c.) Write an N - e proof to verify that the above limit is correct.214 Exercise 4.14.1.3. bk- An Italian math professor confesses he has a pizza-eating problem. He decides to change his usual policy of \"Each minute, I eat all the pizza I see, until it's all gone\Exercise 4.14.1.4. This is For each of the following infinite series, determine if it converges absolutely, converges condition- ally, or diverges. Explain clearly what your reasoning is, citing any tests you use. 18 a.) ( -2) " +n2 n! n=0 b.) E Fn n=0 c.) En=o(-1)" Fn d.) Lin=o V 2 n 3 +n+1 e.) 2n+1 n=0 f.) > n 2n+1 n=0 g. ) Z en ean + 1 n=0Exercise 4.14.2.1. mall Define the following recursive sequence: ao = 2 an+1 = an 2n a.) Compute the first few values of the sequence an. Fill them in the table below: n 0 1 2 3 4 5 an b.) Define AN to be the sequence of partial sums of an. Find the first few values of the sequence AN 4.14. MIXED PRACTICE 215 N 0 1 2 3 4 5 AN c.) Compute the following infinite sum: Can n= 1 How does this quantity relate to your work in part b)?Exercise 4.14.2.2. a.) Formally define what it means for a sequence an to converge to a limit L. b.) Consider the sequence an = 3n+1 What is lim an? n-too c.) Write an N - e proof of your claim in part b. Exercise 4.14.2.3. Consider the following infinite series; 1 1! 2! 3! 4! 51 + ... a.) Apply the Divergence Test/No Hope Test to the above series. What does it tell you about its convergence or divergence? b.) Apply the Alternating Series Test to the above series. What does it tell you about its convergence or divergence? c.) Apply the Ratio Test to the above series. What does it tell you about its convergence or divergence?Exercise 4.14.2.4. bi: Consider the sequence given by the following recurrence relation: 110:0 an=an_1+3n23n+l a.) Write out the rst ve terms of an. CHAPTER 4. SEQUENCES AND SERIES: COMMAS AND PL US SIGNS RUN AMOK b.) Find an explicit formula for an. c.) Does 220:0 an converge or diverge? Explain Why, clearly indicating any tests you use in the process