Chapter 1 Limit 1.1 Limit of Sequence A sequence is an infinite list 1,2,…,n The n-th term of the sequence is an, and n is the index of the term. In this course. we will always assume that all the terms are real numbers. Here are some examples n yn=2:2,2,2 1 1 2’’n 1.-1.1 Un sin n in 1. sin 2. sin 3 Sin n Note that the index does not have to start from 1. For example, the sequence Un actually starts from n=0 (or any even integer). Moreover, a sequence does not have to be given by a formula. For example, the decimal expansions of T give a sequence n:3,3.1,3.14,3.141,3.1415,3.14159,3.141592 If n is the number of digits after the decimal point, then the sequence wn starts at n=0 Now we look at the trend of the examples above as n gets bigger. We find that n gets bigger and can become as big as we want. On the other hand, yn remains constant,zn gets smaller and can become as small as we want. This means that Un approaches 2 and zn approaches 0. Moreover, un and Un jump around and do not approach anything. Finally, wn is equal to T up to the n-th decimal place, and therefore approaches TChapter 1 Limit 1.1 Limit of Sequence A sequence is an infinite list x1, x2, . . . , xn, . . . . The n-th term of the sequence is xn, and n is the index of the term. In this course, we will always assume that all the terms are real numbers. Here are some examples xn = n: 1, 2, 3, . . . , n, . . . ; yn = 2: 2, 2, 2, . . . , 2, . . . ; zn = 1 n : 1, 1 2 , . . . , 1 n , . . . ; un = (−1)n : 1, −1, 1, . . . , (−1)n , . . . ; vn = sin n: sin 1, sin 2, sin 3, . . . , sin n, . . . . Note that the index does not have to start from 1. For example, the sequence vn actually starts from n = 0 (or any even integer). Moreover, a sequence does not have to be given by a formula. For example, the decimal expansions of π give a sequence wn : 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, . . . . If n is the number of digits after the decimal point, then the sequence wn starts at n = 0. Now we look at the trend of the examples above as n gets bigger. We find that xn gets bigger and can become as big as we want. On the other hand, yn remains constant, zn gets smaller and can become as small as we want. This means that yn approaches 2 and zn approaches 0. Moreover, un and vn jump around and do not approach anything. Finally, wn is equal to π up to the n-th decimal place, and therefore approaches π. 7