Chapter2

Chapter2 - 2 Sequences and series We will first deal with...

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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 non-zero number B , eventually all the terms b n will necessarily be non-zero, 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 non-zero. 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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Chapter2 - 2 Sequences and series We will first deal with...

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