10.5 Harmonic Series and p-Series

The Harmonic Series Diverges

The harmonic series is ∑(1/n) from n=1 to ∞. Even though the terms 1/n → 0, the series diverges to infinity. This is a classic surprise: terms shrinking to zero does NOT guarantee convergence. It's often proven using the integral test, comparing it to ∫(1/x)dx, which diverges.

p-Series Test

A p-series has the form ∑(1/n^p) from n=1 to ∞. The rule: if p > 1, the series converges; if p ≤ 1, it diverges. The harmonic series is the special case p = 1, which diverges. This test is a quick way to check convergence without integration.

Using p-Series for Comparison

Because p-series convergence is well known, they're perfect benchmark series for the Direct Comparison Test or Limit Comparison Test. For example, ∑(1/(n²+1)) converges because it behaves like ∑(1/n²), a convergent p-series (p=2 > 1).

Key terms

Harmonic series

∑(1/n) from n=1 to ∞; diverges even though terms approach 0.

p-series

A series of the form ∑(1/n^p); converges if p > 1, diverges if p ≤ 1.

p-series test conclusion for p = 1

This is the harmonic series, which diverges.

Why does terms → 0 not guarantee convergence?

The harmonic series is a counterexample: terms shrink to 0 but the sum still diverges to infinity.

Common use of p-series

Used as benchmark/comparison series in the Direct Comparison Test and Limit Comparison Test.

Try an AP-style question

For which p does Σ 1/nᵖ converge?

1. p > 1
2. p < 1
3. p ≥ 1
4. p > 0