A Purely Geometric Explanation of Integration Around a Pole

One of the central tenets of complex analysis is the residue theorem, which is a result of the fact that contour integrals of all positive integer powers of 1 over z around a pole vanish, except when the power is 1. Is there a better reason for this aside from some seemingly coincidental cancelling out? This capsule shows that it is the same reason that explains why real p-series change from divergent to convergent at p=1.