# Inequality for Harmonic Sum ## Illustration \begin{align*} 1 &+ \underbrace{\frac{1}{2} + \frac{1}{3}} + \underbrace{\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}} + \underbrace{\frac{1}{8} + \frac{1}{9} + \frac{1}{10}} \\ \\ \le 1 &+ \underbrace{\frac{1}{2} + \frac{1}{2}} + \underbrace{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} + \underbrace{\frac{1}{8} + \frac{1}{8} + \frac{1}{8}} \\ \\ \le 1 &+ \underbrace{\frac{1}{2} + \frac{1}{2}}_{=1} + \underbrace{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}_{=1} + \underbrace{\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}}_{=1} \\ \\ \le 1 &+ \log_2(10). \end{align*} ## Formula \[ 1 + \frac{1}{2} + \frac{1}{2} + \dots + \frac{1}{n} \le 1 + \log_2(n). \]