Why is a Telescoping Series Called Telescoping?Date: 03/29/2008 at 18:48:26 From: Bill Subject: Why is telescoping series called telescoping Why is a telescoping series called telescoping? It's more of a point of fact--I understand the concept but I'm interested in why it's called a telescoping series. Date: 03/29/2008 at 21:38:44 From: Doctor Peterson Subject: Re: Why is telescoping series called telescoping Hi, Bill. Have you ever seen an old-fashioned telescope made of two or more tubes that slide inside one another, so you can compactly store it? It's sort of like this: o------------o o------------o o------------o | o------------o| o------------o | o------------o | | | --> | | o------------o | o------------o | o------------o | o------------o| o------------o o------------o A telescoping series similarly lets each term "slide" into the next, "collapsing" the whole thing down to a very simple sum. For a simple example, (1 - 2) + (2 - 3) + (3 - 4) + ... + (n - (n+1)) collapses to 1 + (-2 + 2) + (-3 + 3) + ... + (-n + n) + -(n+1) or 1 - (n+1) = -n The process can be shown nicely this way: (1 - 2) + (2 - 3) + (3 - 4) + ... + (n - (n+1) --------------------------- 1 + 0 + 0 + ... - (n+1) Perhaps that configuration is what prompts the name. It does look a lot like a telescope or tripod leg being collapsed. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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