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.
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.
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.