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Geometric Series

If arithmetic series are the burgers of restaurant menus, then geometric series are the hotdogs. They aren't sold in every restaurant, but we all reminisce about our 6-year-old days when we bite into a diner dog.

It would be neat if a geometric series was made from adding up circles, pentagons, and Pac-Man shaped polygons called nomnomagons. But they're much simpler. A geometric series is a series whose terms form a geometric sequence.

Sample Problem

The series

is a geometric series because

is a geometric sequence. To get from one term to the next in a geometric sequence you must multiply by the same number (called the common ratio) each time; in this example the common ratio is ½.

Sample Problem

The series

is not a geometric series because its terms do not comprise a geometric sequence.

There are two magic formulas we need to know, and know really well, for dealing with geometric series:

  • the formula for the sum of a finite geometric series
      
  • the formula for the sum of an infinite geometric series.

A finite geometric series is the same thing as a partial sum of an infinite geometric series. This means, if we can sum finite geometric series, we'll be able to find partial sums of infinite geometric series.

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