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Remark 121 It would be a mistake to think that every sequence is given by a rule Far from it But many sequences do come from rules, and it is always interesting to determine what that rule is EXAMPLE 122 How does the sequence 1, 2, 3, 4, differ from the sequence in Example 121 Solution: This new sequence has the same values as the sequence in Example 121 But they occur in a different order Since a sequence is by de nition an ordered list, we conclude that this is a different sequence from that in Example 121 Insight: Occasionally it will prove useful to begin a sequence with an index different from 1 An example is {3 j 5} j=4 This denotes the sequence 7, 10, 13, 16, Generally, the main reason for studying a sequence is to understand whether or not it tends to some limit Before giving a precise de nition of limit, let us apply our intuition to some examples EXAMPLE 123 Discuss whether the sequence a j = 2 j which can also be written as {2 j } j=1 tends to a limiting value Solution: If we write the sequence out as 1 1 1 , , , 2 4 8

have selected for renumbering However, the document page number information is still displayed in the small Navigation window below the Document pane

then we see that the terms become (and remain) arbitrarily close to zero It seems plausible to say that the sequence tends to zero This intuitive notion is displayed in Fig 122

i, ii, iii Use this style to display the selected pages as lowercase Roman numerals I, II, III Use this style to display the selected pages as uppercase Roman numerals a, b, c Use this style to display the selected pages as lowercase letters A, B, C Use this style to display the selected pages as capital letters

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EXAMPLE 124 Does the sequence a j = ( 1) j tend to a limit Solution: We may write this sequence out as 1, 1, 1, 1, The sequence does not seem to tend to any limit: half of the time the value is 1 and the other half of the time the value is 1 The sequence does not become and remain close to a single value Therefore we say that it has no limit Refer to Fig 123

6 Enter text in the Prefix window to add a prefix to the page number 7 Click OK to renumber the selected pages

EXAMPLE 125 Does the sequence a j = j 3 tend to a limit Solution: This sequence may be written out as 1, 8, 27, 64, The sequence takes values which are larger and larger, without any bound The sequence tends to no limit Refer to Fig 124

You can add pages to a PDF document at any time by using the Insert Pages command, as discussed previously in this chapter You can append a PDF document by adding existing PDF documents, images, and other supported formats to the document To add existing files to a PDF document, follow these steps: 1 Open the PDF document to which you want to add files If desired, you can open more than one PDF document to combine the open PDF documents with other file types 2 Choose Create PDF | From Multiple Files to open the Create PDF Document From Multiple Files dialog box 3 Make sure the Include All Open PDF documents option is selected 4 Click Browse and choose the file(s) you want to add to the document

EXAMPLE 126 A quantity of radioactive material decays At the beginning of each week there is half as much as there was the previous week The initial quantity is 5 grams Use sequence notation to express the amount of material at the beginning of the jth week Solution: The amount of radioactive material at the beginning of the second week is 5/2 (half as much as the initial amount at the beginning of the rst week) The amount at the beginning of the third week is 5/4 The amount at the beginning of the fourth week is 5/8 And so forth As a result, according to the description, the amount of material at the start of the jth week is aj = 5 1 2

You can combine two PDF documents by opening one PDF document and choosing Document | Pages | Insert Select the PDF file and choose the proper options to insert the new PDF file before or after the current PDF file

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