Determine the values of p for which the series ? ln n np n = 2 converges. (if the answer is an interval, use interval notation. if the answer is a finite set, use set notation.)

I assume the series in question is



Compare the series to the larger integral,



Integrate by parts taking,




So the integral is equivalent to




The constant term explodes when , so we require . The remaining limit term can be written as



where the last equality is obtained by applying L'Hopital's rule once.

Now, the limit will only exist if , so this is the condition required for the integral to converge, and hence for the series to converge.

Two because two of one thing is two.