then is a continuous, positive, decreasing, function. Further, see example . . . ,

is convergent for and divergent for .
Thus,

is divergent for .

.

To analyze the convergence/ divergence of this series, we can proceed as follows: Since
,

and the series is divergent (p=1 for the p-series), by comparison test, is also divergent. We could directly apply the integral test with . As

we can conclude that the series

is divergent.