\sum_{k=1}^{n+1}{\frac{1}{k}} = \sum_{k=1}^{n}{\frac{1}{k}}+ \frac {1}{n+1} \geq \frac{2n}{n+1} + \frac{1}{n+1} = \frac{2n+1}{n+1} \geq \frac{2n+2}{n+2}