[tex]1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} = \\ \\ 1 + \frac{1}{2} = \frac{1}{1} + \frac{1}{2} = \frac{1 \times 2}{1 \times 2} + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \\ \\ \frac{3}{2} + \frac{1}{3} = \frac{3 \times 3}{2 \times 3} + \frac{1 \times 2}{3 \times 2} = \frac{9}{6} + \frac{2}{6} = \frac{9 + 2}{6} = \frac{11}{6} \\ \\ \frac{11}{6} + \frac{1}{4} = \frac{11 \times 2}{6 \times 2} + \frac{1 \times 3}{4 \times 3} = \frac{22}{12} + \frac{3}{12} = \frac{22 + 3}{12} = \frac{25}{12} \\ \\ \frac{25}{12} + \frac{1}{2} = \frac{25}{12} + \frac{1 \times 6}{2 \times 6} = \frac{25}{12} + \frac{6}{12} = \frac{25 + 6}{12} = \frac{31}{12} \\ \\ \frac{31}{12} + \frac{2}{3} = \frac{31}{12} + \frac{2 \times 4}{3 \times 4} = \frac{31}{12} + \frac{8}{12} = \frac{31 + 8}{12} = \frac{39}{12} = \frac{13}{4} \\ \\ \frac{13}{4} + \frac{3}{4} = \frac{13 + 3}{4} = \frac{16}{4} = 4 [/tex]
