Example 3 The geometricseries has all non-negativeterms.The sequenceof partialsums
lookslike this:
Intuitively, we can see that thereis no boundon the sequenceof partialsumsand so the seriesdiverges.
This is confirmedby the fact that the ratio of the series, , tells us that the geometricseriesdoesnot
converge.
Anotherexampleof an importantseriesis the p-series:
, wherep> 0.
Thep-seriesmay look like a harmonicseries,but it will convergefor certainvaluesofp.
TheoremThep-series convergesforp> 1 and divergesfor 0 <pā¤ 1.Example 4 Determineif convergesor diverges.
Solution