Comparison turns the behavior of a difficult series into a question about a familiar bound.
For a positive decreasing function, the integral from 1 to N+1 lies below the sum from 1 to N. The integral of x⁻ᵖ has a finite infinite tail exactly for p>1; complementary upper bounds give convergence in that case. For p≤1, the terms dominate the divergent harmonic series or the integral diverges.
Upper bounds prove convergence and lower bounds prove divergence when the comparison series has the corresponding known behavior.