The convergenceor divergenceof a geometricseriesdependsonr.
TheoremSupposethat the geometricseries has ratior. 1. The geometricseriesconvergesif
|r| < 1 and the sum of the seriesis. 2. The geometricseriesdivergesif |r| ≥ 1.Example 5
Determineif the series convergesor diverges.If it converges,find
the sum of the series.
Solution
The seriesis a geometricseriesthat can be writtenas. Thena= 7 and the ratio.
Because , the seriesconverges.The sum of the seriesis.
Example 6
Determineif the series convergesor diverges.If it converges,find the sum of the series.
Solution
The seriesis a geometricserieswitha= 1 and the ratior= 9. Because|9|>1,the seriesdiverges.
Example 7
Determineif convergesor diverges.If it converges,find the sum of
the series.
Solution
If we rewritethe seriesin termsof powersofk, the serieslookslike this:
.It lookslike a geometricserieswitha= 3 and .Since , the seriesconverges.
However, if we writethe definitionof a geometricseriesfora= 3 and , the serieslookslike this: