EXAMPLE 2 For what values of p is the series [infinity] 1 np n = 1 convergent? SOLUTION If p < 0, then lim n → [infinity] (1/np) = [infinity]. If p = 0, then lim n → [infinity] (1/np) = . In either case, lim n → [infinity] (1/np) ≠ 0, so the given series diverges by the Test for Divergence. If p > 0, then the function f(x) = 1/xp is clearly continuous, positive, and decreasing on [1, [infinity]). Previously, we found that [infinity] 1 xp dx 1 converges if p > and diverges if p ≤ . It follows from the Integral Test that the series Σ 1/np converges if p > and diverges if 0 < p ≤ . (For p = 1, the series is the harmonic series discussed in this example.)