> [!NOTE] Definition > The $n$th *harmonic number* $H_{n}$ is given by the sum $H_{n} = \sum_{k=1}^{n} \frac{1}{k} $ # Properties Lemma ($H_{n}$ is properly divergent) >Proof. We can prove that it [[Convergent Real Series|diverges]] using >- [[Terms of Convergent Series Tend to Zero]]; >- [[Comparison Test for Series With Non-Negative Terms]]; >- [[Integral test for convergence of series of non-negative decreasing function]]; >- [[Series with Non-Negative Terms Converges Iff Partial Sums Are Bounded Above]] # Applications - [[Alternating Harmonic Series]].