An interval can be infinitely long while the total area above it is finite. The decay rate decides.
WHAT HAPPENS IF…Compare p just above 1 with p = 1.
Interactive exploration
Preparing your experiment…
Connect the picture to the idea∫1∞x−pdx=p−11(p>1)
For p ≠ 1, the integral to R is (R^(1−p) − 1)/(1−p). If p > 1 the power tends to zero. At p = 1 the integral is ln R, which grows without bound; smaller p also diverges.
A detail that matters
A function tending to zero is not enough to make its improper integral converge.