The Book of Revelation For Dummies
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You don’t see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Consider the following series:

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To see that this is a telescoping series, you have to use the partial fractions technique to rewrite

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All these terms now collapse, or telescope. The 1/2s cancel, the 1/3s cancel, the 1/4s cancel, and so on. All that’s left is the first term, 1 (actually, it’s only half a term), and the last half-term,

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and thus the sum converges to 1 – 0, or 1.

You can write each term in a telescoping series as the difference of two half-terms — call them h-terms. You can then write the telescoping series as

image3.png

Here's the telescoping series rule: A telescoping series of the above form converges if

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then the series diverges.

This rule, like the rule for geometric series, lets you determine what number a convergent telescoping series converges to.

About This Article

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About the book authors:

Richard Wagner is the inventor and architect of the award-winning NetObjects ScriptBuilder.

Richard Mansfield has authored more than 30 computer books since 1982.

Larry R. Helyer is a professor of biblical studies at Taylor University in Upland, Indiana. He received his doctorate in New Testament from Fuller Theological Seminary in Pasadena, California. He pastored Baptist churches in Portland, Oregon, and Sun Valley, California, before moving to the Midwest and teaching biblical studies at Taylor University for 28 years. Helyer is the author of two books, Yesterday, Today, and Forever: The Continuing Relevance of the Old Testament and Exploring Jewish Literature of the Second Temple Period: A Guide for New Testament Students. He's also written numerous journal and dictionary articles on biblical and theological subjects and a book on New Testament theology. He was the initial translator of "2 Samuel" for the Holman Christian Standard Bible.

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