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SalasSV_10_02

# SalasSV_10_02 - 590 CHAPTER 10 SEQUENCES INDETERMINATE...

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590 ± CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS ± 10.2 SEQUENCES OF REAL NUMBERS To this point we have considered sequences only in a peripheral manner. Here we focus on them. What is a sequence of real numbers? DEFINITION 10.2.1 SEQUENCE OF REAL NUMBERS A sequence of real numbers is a real-valued function deFned on the set of positive integers. You may Fnd this deFnition somewhat surprising, but in a moment you will see that it makes sense. Suppose we have a sequence of real numbers a 1 , a 2 , a 3 , ... , a n , . What is a 1 ? It is the image of 1. What is a 2 ? It is the image of 2. What is a 3 ?Itisthe image of 3. In general, a n is the image of n . By convention, a 1 is called the Frst term of the sequence, a 2 the second term , and so on. More generally, a n , the term with index n , is called the n th term . Sequences can be deFned by giving the law of formation. ±or example: a n = 1 n is the sequence 1, 1 2 , 1 3 , 1 4 , ; b n = n n + 1 is the sequence 1 2 , 2 3 , 3 4 , 4 5 , ; c n = n 2 is the sequence 1, 4, 9, 16, . It’s like deFning f by giving f ( x ). Sequences can be multiplied by constants; they can be added, subtracted, and multiplied. ±rom a 1 , a 2 , a 3 , , a n , and b 1 , b 2 , b 3 , , b n , we can form the scalar product sequence : α a 1 , α a 2 , α a 3 , , α a n , , the sum sequence : a 1 + b 1 , a 2 + b 2 , a 3 + b 3 , , a n + b n , , the difference sequence : a 1 b 1 , a 2 b 2 , a 3 b 3 , , a n b n , , the product sequence : a 1 b 1 , a 2 b 2 , a 3 b 3 , , a n b n , . If the b i s are all different from zero, we can form the reciprocal sequence : 1 b 1 , 1 b 2 , 1 b 3 , , 1 b n , , the quotient sequence : a 1 b 1 , a 2 b 2 , a 3 b 3 , , a n b n , . The range of a sequence is the set of values taken on by the sequence. The range of the sequence a 1 , a 2 , a 3 , , a n , is the set { a n : n = 1, 2, 3, } .

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10.2 SEQUENCES OF REAL NUMBERS ± 591 Thus the range of the sequence 0, 1, 0, 1, 0, 1, 0, 1, ... made up of alternating zeros and ones is the set { 0, 1 } . Can you Fnd a law of formation a n for this sequence? The range of the sequence 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, is the set of integers. Boundedness and unboundedness for sequences are what they are for other real- valued functions. Thus the sequence a n = 2 n is bounded below (with greatest lower bound 2) but unbounded above. The sequence b n = 2 n is bounded. It is bounded below with greatest lower bound 0, and it is bounded above with least upper bound 1 2 . Many of the sequences we work with have some regularity. They either have an upward tendency or they have a downward tendency. The following terminology is standard. The sequence { a n } is said to be increasing if a n < a n + 1 for all n , nondecreasing if a n a n + 1 for all n , decreasing if a n > a n + 1 for all n , nonincreasing if a n
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SalasSV_10_02 - 590 CHAPTER 10 SEQUENCES INDETERMINATE...

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