Theorem 4.2.11: p Series
The series
is called a p Series.
 if p > 1 the pseries converges
 if p 1 the pseries diverges
Examples 4.2.12:  

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 ^{1p} 1, hence the righthand series diverges
 if 1 < p then 2 ^{1p} < 1, hence the righthand series converges
Now the result follows from the Cauchy Condensation test .