Solveforsn by multiplyingboth sidesof the last equationby :
Now we find the limit of both sides:
The sum of the infiniteseriesis and so the seriesconverges.
GeometricSeries
Thegeometricseriesis a specialkind of infiniteserieswhoseconvergenceor divergenceis basedon a
certainnumberassociatedwith the series.
GeometricSeriesA geometricseriesis an infiniteserieswrittenasa + ar + ar^2 + ar^3 +...+ar{i-1}+.... Insigmanotation,a geometricseriesis writtenas. The numberris theratioof the series.Example 4
Hereare someexamplesof geometricseries.
GeometricSeries
| a |r
1
51 + 3 + 3^2 + 3^3 +...+ 3k-1+...^13