Discuss the convergence of .
This type of series is called a p-series. The idea of this problem is to find which values of p give us a convergent series and which values of p give us a divergent series. When we have that info, we can find the convergence or divergence of any p-series without having to calculate hundreds of different integrals separately, simply by knowing the value of p. Use the integral test to determine convergence and divergence.
First, if the integral yields
So, when the integral diverges.
If the integral converges because as the integral goes to .
If the integral diverges because as the integral goes to .
So, by the integral test, the series
converges when and diverges otherwise, for any real p.