Math Problem Solver

\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \frac{1}{22} = \frac{83711}{55440} = 1.5099\overline{386724} \]

The sum can be computed by rewriting each term over a common denominator, where the least common denominator of \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{6}\), \(\frac{1}{8}\), \(\frac{1}{10}\), \(\frac{1}{12}\), \(\frac{1}{14}\), \(\frac{1}{16}\), \(\frac{1}{18}\), \(\frac{1}{20}\), and \(\frac{1}{22}\) is \(55440\). \[ \begin{aligned} \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \frac{1}{22} &= \frac{1 \cdot 27720}{2 \cdot 27720} + \frac{1 \cdot 13860}{4 \cdot 13860} + \frac{1 \cdot 9240}{6 \cdot 9240} + \frac{1 \cdot 6930}{8 \cdot 6930} + \frac{1 \cdot 5544}{10 \cdot 5544} + \frac{1 \cdot 4620}{12 \cdot 4620} + \frac{1 \cdot 3960}{14 \cdot 3960} + \frac{1 \cdot 3465}{16 \cdot 3465} + \frac{1 \cdot 3080}{18 \cdot 3080} + \frac{1 \cdot 2772}{20 \cdot 2772} + \frac{1 \cdot 2520}{22 \cdot 2520} \\ &= \frac{27720 + 13860 + 9240 + 6930 + 5544 + 4620 + 3960 + 3465 + 3080 + 2772 + 2520}{55440} \\ &= \frac{83711}{55440} \end{aligned} \]