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the way it should be: the convergence or divergence of a series depends only on its tail Now we look at some examples that illustrate when the ratio test gives convergence, when it gives divergence, and when it gives no information EXAMPLE 1340 Apply the ratio test to the series

Alternatively, you can right-click (Windows) or CTRL-click (Macintosh) within the Pages tab to open a Context menu The Pages tab context menu is a watered-down version of the Pages Options menu However, you may prefer to use the Context menu when you need to access its commands quickly You can access the complete version of the Pages Options menu by selecting a thumbnail and then right-clicking on a Windows machine, or CTRL-clicking on a Macintosh machine

j=1

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When you create a PDF document in an authoring application or using the Adobe PDFMaker plug in, you can modify a setting to embed thumbnails Thumbnails increase the file size of the published PDF document by approximately 3K or more per thumbnail for a page with images; thumbnails for pages with text are only slightly smaller The actual size of the thumbnail will vary depending on the amount of graphics in the page from which the thumbnail is generated When you create a document without embedded thumbnails, Acrobat creates them dynamically

2j j!

when you open the Pages tab This can take several seconds for a large document You can choose to embed thumbnails to circumvent the redraw every time you open the palette You can easily remove embedded thumbnails in the future if desired To embed all thumbnails within a document, follow these steps: 1 Open the Navigation pane and click Pages to open the Pages tab 2 Open the Pages Options menu and choose Embed All Page Thumbnails Alternatively, you can choose the Embed All Page Thumbnails command from the Pages tab Context menu To remove embedded thumbnails from a document, do the following: 1 Open the Navigation pane and click Pages to access the Pages tab 2 Click Options to open the Pages Options menu and choose Remove Embedded Page Thumbnails Alternatively, you can right-click (Windows) or CTRL-click (Macintosh) and choose Remove Embedded Page Thumbnails from the Context menu After you remove thumbnails from a document, Acrobat will create them on the fly whenever the Pages tab is opened As mentioned previously, this may take considerable time with a lengthy document As a rule, you should remove embedded thumbnails only when the file size of the published PDF document is a factor

Solution: Recall that j! = j ( j 1) ( j 2) 3 2 1 Observe that c j = 2 j / j! Then c j+1 2 j+1 /( j + 1)! = cj 2 j / j! = 2 j +1

When you open the Pages tab, all thumbnails are displayed at default size Unless you resize the width of the Navigation pane, the thumbnails are displayed in a single column If you prefer, you can display more thumbnails in the Navigation pane by choosing Reduce Page Thumbnails from the Pages Options menu After you choose this command, thumbnails are displayed at reduced magnification To return thumbnails to their original size, choose Enlarge Page Thumbnails from the Pages Options menu You can use the Reduce Page Thumbnails or Enlarge Page Thumbnails command more than once to arrive at the optimum thumbnail size for your working preference

As j , the limit of this expression is 0 Therefore in this example the limit L exists and equals 0 By Part (1) of the ratio test, we may conclude that the series converges absolutely

As you have already learned, when you convert a document into PDF format, navigation devices in the form of thumbnails and bookmarks are created for you You can, however, exceed the limitations of these rudimentary navigation devices by creating links When you create a link, you create a hotspot in the document that changes the user s cursor into a pointing hand when a

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A black border surrounds each thumbnail You can change page magnification by clicking and dragging the black rectangle at the thumbnail s lower-right corner After changing the page magnification, a red rectangle surrounds the current page view from which you can click and drag any borderline to scroll to a different part of the document

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