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# chapter4 - Chapter 4 Sequences and Series 4.1 Innite...

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Chapter 4. Sequences and Series 4.1 Infinite sequences An infinite sequence (or sequence) of real numbers is an infinite succession of numbers which is usually given by some rule. We shall denote an infinite sequence by a 1 , a 2 , a 3 , · · · , a n , · · · , and we shall often write the sequence as { a n } ; and for each n , the number a n is called a term of the sequence.

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2 MA1505 Chapter 4. Sequences and Series 4.1.1 Example (i) The sequence 0 , 1 , 2 , · · · , n - 1 , · · · is defined by the rule a n = n - 1. (ii) The sequence 1 , 1 2 , 1 3 , · · · , 1 n , · · · is defined by a n = 1 n . (iii) If a n = ( - 1) n +1 ( 1 n ), the sequence is 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , · · · . (iv) If a n = n - 1 n , the sequence is 0 , 1 2 , 2 3 , 3 4 , 4 5 , · · · . 2
3 MA1505 Chapter 4. Sequences and Series (v) If a n = ( - 1) n +1 , the sequence is 1 , - 1 , 1 , - 1 , · · · . (vi) If a n = 3, the sequence is 3 , 3 , 3 , · · · 4.1.2 Limits of sequences A number L is called the limit of a sequence { a n } , if for sufficiently large n , we can get a n as close as we want to a number L . We write lim n →∞ a n = L , or simply, a n L , Note that the limit of a sequence { a n } is unique. 4.1.3 Convergent and divergent Not all sequences have limits. 3

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4 MA1505 Chapter 4. Sequences and Series If { a n } has a limit, we say the sequence is convergent and { a n } converges to L . If { a n } does not have a limit, we say { a n } is diver- gent. 4.1.4 Example (i) 0 , 1 , 2 , 3 , · · · is divergent. (ii) 1 , 1 2 , 1 3 , · · · , 1 n , · · · is convergent, its limit is 0. (iii) 0 , 1 2 , 2 3 , 3 4 , 4 5 , · · · converges to 1. (since lim n →∞ n - 1 n = 1. See example 4.1.6 (ii).) (iv) 1 , - 1 , 1 , - 1 , · · · is divergent. (v) 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , · · · . converges to 0. (vi) If c is any real number, c, c, c, · · · clearly con- 4
5 MA1505 Chapter 4. Sequences and Series verges to c . Such a sequence is called a constant sequence. 4.1.5 Some Rules on Limits Let lim n →∞ a n = A , and lim n →∞ b n = B , with A and B real numbers. (1) Sum rule: lim n →∞ ( a n + b n ) = A + B . (2) Difference rule: lim n →∞ ( a n - b n ) = A - B . (3) Product rule: lim n →∞ ( a n b n ) = AB . (4) Quotient rule: lim n →∞ a n b n = A B , if B 6 = 0. Using the above rules, we obtain: 4.1.6 Example (i) lim n →∞ ( - 1 n ) = ( - 1) lim n →∞ 1 n = - 1(0) = 0. 5

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6 MA1505 Chapter 4. Sequences and Series (ii) lim n →∞ n - 1 n = lim n →∞ (1 - 1 n ) = lim n →∞ 1 - lim n →∞ 1 n = 1 - 0 = 1. (iii) lim n →∞ 5 n 2 = 5 lim n →∞ 1 n · lim n →∞ 1 n = 5 · 0 · 0 = 0. (iv) lim n →∞ 4 - 7 n 6 n 6 + 3 = lim n →∞ 4 n 6 - 7 1 + 3 n 6 = 0 - 7 1 + 0 = - 7. (Note that all the four rules in (4.1.5) are used in this example.) 4.1.7 Sequence and function Let { a n } be a sequence. Suppose there is a function f ( x ) such that a n = f ( n ). If lim x →∞ f ( x ) = L , then lim n →∞ a n = L. 6
7 MA1505 Chapter 4. Sequences and Series 4.1.8 Example Consider the sequence { a n } where a n = n + 1 - n . Then a n = f ( n ) where f ( x ) is the function x + 1 - x . So lim n →∞ a n = lim x →∞ ( x + 1 - x ) = lim x →∞ 1 x + 1 + x = 0 .

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