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Unformatted text preview: Sequences and their limits Notes for MATH 138 The limit idea For the purposes of calculus, a sequence is simply a list of numbers x 1 ,x 2 ,x 3 ,...,x n ,... that goes on indefinitely. The numbers in the sequence are usually called terms , so that x 1 is the first term, x 2 is the second term, and the entry x n in the general n th position is the n th term , naturally. The subscript n = 1 , 2 , 3 ,... that marks the position of the terms will sometimes be called the index . We shall deal only with real sequences, namely those whose terms are real numbers. Here are some examples of sequences. • the sequence of positive integers: 1 , 2 , 3 ,...,n,... • the sequence of primes in their natural order: 2 , 3 , 5 , 7 , 11 ,... • the decimal sequence that estimates 1 / 3 : . 3 ,. 33 ,. 333 ,. 3333 ,. 33333 ,... • a binary sequence: , 1 , , 1 , , 1 ,... • the zero sequence: , , , ,... • a geometric sequence: 1 ,r,r 2 ,r 3 ,...,r n ,... • a sequence that alternates in sign: 1 2 , 1 3 , 1 4 ,..., ( 1) n n ,... • a constant sequence: 5 , 5 , 5 , 5 , 5 ,... • an increasing sequence: 1 2 , 2 3 , 3 4 , 4 5 ..., n n +1 ,... • a decreasing sequence: 1 , 1 2 , 1 3 , 1 4 ,..., 1 n ,... • a sequence used to estimate e : ( 3 2 ) 2 , ( 4 3 ) 3 , ( 5 4 ) 4 ..., ( n +1 n ) n ... 1 • a seemingly random sequence: sin1 , sin2 , sin3 ,..., sin n,... • the sequence of decimals that approximates π : 3 , 3 . 1 , 3 . 14 , 3 . 141 , 3 . 1415 , 3 . 14159 , 3 . 141592 , 3 . 1415926 , 3 . 14159265 ,... • a sequence that lists all fractions between and 1 , written in their lowest form, in groups of increasing denominator with increasing numerator in each group: 1 2 , 1 3 , 2 3 , 1 4 , 3 4 , 1 5 , 2 5 , 3 5 , 4 5 , 1 6 , 5 6 , 1 7 , 2 7 , 3 7 , 4 7 , 5 7 , 6 7 , 1 8 , 3 8 , 5 8 , 7 8 , 1 9 , 2 9 , 4 9 , 5 9 ,... It is plain to see that the possibilities for sequences are endless. Ways to prescribe a sequence A sequence is prescribed by making clear what its n th term is supposed to be. We can use a long list to indicate a pattern, but shorter notations such as { x n } ∞ n =1 , or more briefly { x n } , or even the unadorned x n are suitable as well. For some sequences it is possible to give a simple formula for the n th term as a function of the index n . For example, the n th term of the sequence 1 , 1 / 2 , 1 / 3 , 1 / 4 ,... is x n = 1 /n . For other sequences, such as the sequence of primes or the sequence for the decimal expansion of π , a clean formula for the n th term is not available. Nevertheless, the entry in the n th position remains uniquely specified. At times the sequence { x n } is given, not by a direct formula for the n th term, but rather recursively . To specify a sequence recursively, you state explicitly what one or more of the beginning terms are, and then you give a formula for the general entry in terms of its preceding terms. Here is an example of a famous sequence that is defined recursively. Let f = 1 ,f 1 = 1 , and for indices...
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This note was uploaded on 03/16/2011 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.
 Winter '07
 Anoymous
 Calculus, Limits

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