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Unformatted text preview: 2 Sequences and series We will first deal with sequences, and then study infinite series in terms of the associated sequence of partial sums. 2.1 Sequences By a sequence , we will mean a collection of numbers a 1 , a 2 , a 3 , . . . , a n , a n +1 , . . . , which is indexed by the set N of natural numbers. We will often denote it simply as { a n } . A simple example to keep in mind is given by a n = 1 n , which appears to decrease towards zero as n gets larger and larger. In this case we would like to have 0 declared as the limit of the sequence . A quick example of a sequence which does not tend to any limit is given by the sequence { 1 , 1 , 1 , 1 , . . . } , because it just oscillates between two values; for this sequence, a n = ( 1) n +1 , which is certainly bounded. Definition 2.1 A sequence { a n } is said to converge, i.e., have a limit A , iff for any > there exists N = N ( ) > s.t. for all n N we have  a n A  < . Notation: lim n a n = A , or a n A as n . Two Remarks : (i) It is immediate from the definition that for a sequence { a n } to converge, it is necessary that it be bounded. However, it is not sucient , i.e., { a n } could be bounded without being convergent. Indeed, look at the example a n = ( 1) n +1 considered above. (ii) It is only the tail of the sequence which matters for convergence. Oth erwise put, we can throw away any number of the terms of the sequence occurring at the beginning without upsetting whether or not the later terms bunch up near a limit point. Lemma 2.2 If lim n a n = A and lim n b n = B , then 1 1) lim n ( a n + b n ) = A + B 2) lim n ( ca n ) = cA , for any c R 3) lim n a n b n = AB 4) lim n a n /b n = A/B , if B = 0 . Proof of 1) For any > 0, choose N 1 and N 2 so that for n 1 N 1 , n 2 N 2 ,  a n 1 A  < / 2,  b n B  < / 2. Then for n max { N 1 , N 2 } , we have  a n + b n A B  < 2 + 2 = . Hence A + B is the limit of a n + b n as n . 2 Proofs of the remaining three assertions are similar. For the last one, note that since { b n } converges to a nonzero number B , eventually all the terms b n will necessarily be nonzero, as they will be very close to B . So, in the sequence a n /b n , we will just throw away some of the initial terms when b n = 0, which doesnt affect the limit as the b n occurring in the tail will all be nonzero. 2 Example : We have lim n 1 n = 0. Indeed, given > 0, we may choose an N N such that N > 1 , because N is unbounded. This implies that for n N , we have  1 n  1 N < . Hence { 1 n } converges to 0. 2 Definition A sequence is said to be monotone increasing , denoted a n , if a n +1 a n for all n 1, and monotone decreasing , denoted a n , if a n +1 a n for n 1. We say { a n } is monotone (or monotonic) if it is of one of these two types....
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 Winter '08
 Borodin,A
 Sequences And Series, Candide

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