证明级数 $$\bex 1+\frac{1}{2}-\frac{1}{3} +\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots \eex$$ 发散.
证明: 由 $$\beex \bea |S_{6n}-S_{3n}|&=\sev{\sum_{k=n+1}^{2n}\sex{\frac{1}{3k-2}+\frac{1}{3k-1}-\frac{1}{3k}}}\\ &>\sum_{k=n+1}^{2n} \frac{1}{3k-2}\quad\sex{\frac{1}{3k-1}>\frac{1}{3k}}\\ &>\sum_{k=n+1}^{2n}\frac{1}{3k}\\ &>\frac{1}{3\cdot 2n}\cdot n=\frac{1}{6} \eea \eeex$$ 及 Cauchy 收敛准则即知结论成立.
