> [!NOTE] Definiton (Alternating Harmonic Series) >The alternating harmonic series is the [[Series of Real Sequence|series]]$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} +\dots$ # Properties > [!NOTE] Proposition (Convergence) > Contents >*Proof*. Write $s_{k} = \sum_{n=1}^{k} (-1)^{n+1} \frac{1}{n}.$ Note that $\begin{align} \frac{1}{2n-1} - \frac{1}{2n} &= \frac{1}{2n(2n-1)} \\ & \leq \frac{1}{n^2} \end{align}$So $\begin{align} s_{2k} & \leq \sum_{n=1}^{k} \frac{1}{n^2} \\ &\leq \sum_{n=1}^{\infty} \frac{1}{n^{2}} < \infty \end{align}$also $\begin{align} s_{2k+2} &= s_{2k} + \frac{1}{2k+1} - \frac{1}{2k+2} \\ & \geq s_{2k} \end{align}$so $(s_{2k})$ is increasing and bounded. >so, by [[Monotone Bounded Real Sequence is Convergent|monotone convergence]], $s_{2k} \to l \in \mathbb{R}$ as $k \to \infty$ and $s_{2k+1} = s_{2k} + a_{2k+1} \to \infty$ as $k \to \infty.$ Therefore $s_{k} \to l$ as $k \to l$ since [[If even and odd terms of sequence converge to the same limit then the sequence converges to that limit]]. >*Proof*. Follows from [[Alternating series test]]. > [!NOTE] Lemma (Absolute Divergence) > The alternating harmonic is [[Absolutely Convergent Series|absolutely divergent]]. >Proof. Absolute alternating harmonic series is [[Harmonic Numbers|harmonic series]].