Theorem 4.2.11: p Series
The series is called a p Series.
- if p > 1 the p-series converges
- if p 1 the p-series diverges
If p < 0 then the sequence converges to infinity. Hence, the series diverges by the Divergence Test.
If p > 0 then consider the series
=The right hand series is now a Geometric Series.
- if 0 < p 1 then 2 1-p 1, hence the right-hand series diverges
- if 1 < p then 2 1-p < 1, hence the right-hand series converges
Now the result follows from the Cauchy Condensation test .