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i 2/n 3
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We can expand it to 12/n 3 + 22/n 3 + 32/n 3 + + n 2/n 3 + which simplifies to (12 + 22 + 32 + + n 2 + )/n 3 As n grows endlessly larger, we have Lim (12 + 22 + 32 + + n 2)/n 3
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This limit turns out to be 1/3 Therefore Lim
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i 2/n 3 = 1/3
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Still another example Finally, let s look at an infinite series where we cube a positive integer i and then divide it by the fourth power of another positive integer n Symbolically, we write this as
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i 3/n 4
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When we write this series out, we obtain 13/n 4 + 23/n 4 + 33/n 4 + + n 3/n 4 + which simplifies to (13 + 23 + 33 + + n 3 + )/n 4
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Sequences, Series, and Limits
As n grows endlessly larger, we have Lim (13 + 23 + 33 + + n 3)/n 4
This limit turns out to be 1/4 Therefore Lim
i 3/n 4 = 1/4
i =1
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Figure 19-6 is a graph of the first few elements of an infinite arithmetic sequence If we call the sequence S, then S = s 0, (s 0 + c ), (s 0 + 2c ), (s 0 + 3c ), where s0 is the initial term value and c is a constant Based on the information given in this graph, what is s0 What is c What is the value of the hundredth term in S
6 4 2
Term value 5, 3, 1, 1, 3, 5,
Term number
2 4
Figure 19-6
Illustration for Problem 1
Practice Exercises
2 Does the infinite arithmetic sequence described in Problem 1 converge If so, on what value does it converge If not, why not 3 The general form for an infinite geometric sequence T is T = t0, t 0k, t 0k2, t 0k 3, t 0k 4, where t0 is the initial value and k is the constant of multiplication Calculate, and write down, the first seven terms in an infinite geometric sequence T where t 0 = 2 and k = 4 Does this sequence converge If so, on what value does it converge If not, why not 4 Suppose that in the scenario of Problem 3, we change k from 4 to 1/4 Calculate and list the first seven values of the resulting infinite sequence Does it converge If so, on what value does it converge If not, why not 5 Consider again the sequence we saw earlier in this chapter: B = 0/1, 1/2, 2/3, 3/4, 4/5, , (n 1)/n, We determined that the limit of B, as n grows without end, is Lim (n 1)/n = 1
so we know that B converges Write down the series B+ that we get when we add the elements of B Then write down the first five terms of the sequence B*, which is made up of the partial sums in B+ Does the sequence B* converge If so, to what value does it converge If not, why not 6 Express the following series by writing out the first five terms followed by an ellipsis: S+ =
1/10i
i =1
First, express the terms as fractions Then express them as powers of 10 Then express them as decimal quantities Finally, write down the first five terms in the sequence S* of partial sums 7 Find the following limit if it exists If no limit exists, explain: Lim
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