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### 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/
Associated Topics:
High School Definitions
High School Sequences, Series

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