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It s reasonable to ask, Can we categorize all sequences as either arithmetic or geometric The answer is no! Consider U = 10, 13, 17, 22, 28, 35, 43, This sequence shows a pattern, but it s neither arithmetic nor geometric The difference between the first and second terms is 3, the difference between the second and third terms is 4, the difference
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Sequences, Series, and Limits
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between the third and fourth terms is 5, and so on The difference keeps increasing by 1 for each succeeding pair of terms This is a fairly simple example of a nonarithmetic, nongeometric sequence with an identifiable pattern
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Suppose a particular species of cell undergoes mitosis (splits in two) every half hour, precisely on the half hour We take our first look at a cell culture at 12:59 pm, and find three cells At 1:00 pm, mitosis occurs for all the cells at the same time, and then there are six cells in the culture At 1:30 pm, mitosis occurs again, and we have 12 cells How many cells are there in the culture at 4:01 pm
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There are 3 hours and 2 minutes between 12:59 pm and 4:01 pm This means that mitosis takes place 7 times: at 1:00, 1:30, 2:00, 2:30, 3:00, 3:30, and 4:00 Table 19-1 illustrates the scenario We look at the culture repeatedly at 1 minute past each half hour There are 384 cells at 4:01 pm, just after the mitosis event that occurs at 4:00 pm
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Table 19-1 Cell division as a function of time, assuming mitosis occurs every half hour
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Time 12:59 1:01 1:31 2:01 2:31 3:01 3:31 4:01 Number of cells 3 6 12 24 48 96 192 384
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Limit of a Sequence
A limit is a specific, well-defined quantity that a sequence, series, relation, or function approaches The value of the sequence, series, relation, or function can get arbitrarily close to the limit, but doesn t always reach it
An example Let s look at an infinite sequence A that starts with 1 and then keeps getting smaller For any positive integer n, the nth term is 1/n, so we have
A = 1, 1/2, 1/3, 1/4, 1/5, , 1/n,
Limit of a Sequence
This is a simple example of a special type of sequence called a harmonic sequence In this particular case, the values of the terms approach 0 The hundredth term is 1/100; the thousandth term is 1/1000; the millionth term is 1/1,000,000 If we choose a tiny but positive real number, we can always find a term in the sequence that s closer to 0 than that number But no matter how much time we spend generating terms, we ll never get 0 We say that The limit of 1/n, as n approaches infinity, is 0, and write it as Lim 1/n = 0
Another example Consider the sequence B in which the numerators ascend one by one through the set of natural numbers, while every denominator is equal to the corresponding numerator plus 1 For any positive integer n, the nth term is (n 1)/n, so we have
B = 0/1, 1/2, 2/3, 3/4, 4/5, , (n 1)/n, As n becomes extremely large, the numerator (n 1) gets closer and closer to the denominator, when we consider the difference in proportion to the value of n Therefore Lim (n 1)/n = n/n = 1
Still another example Let s see what happens in a sequence C where every numerator is equal to the square of the term number, while every denominator is equal to twice the term number For any positive integer n, the nth term is n 2/(2n), so we have
C = 1/2, 4/4, 9/6, 16/8, 25/10, 36/12, 49/14, , n 2 / (2n), Note that n 2/(2n) = n /2 This tells us that Lim n 2/(2n) = Lim n /2
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