1. **State the problem:** We need to find the sum of the fractions $\frac{1}{7} + \frac{1}{5} + \frac{1}{2}$ and express the result in its simplest form. 2. **Formula and rules:** To add fractions, we find a common denominator, add the numerators, and simplify the result. 3. **Find the least common denominator (LCD):** The denominators are 7, 5, and 2. The LCD is the least common multiple of 7, 5, and 2. Since 7, 5, and 2 are all prime to each other, the LCD is $7 \times 5 \times 2 = 70$. 4. **Rewrite each fraction with denominator 70:** $$\frac{1}{7} = \frac{1 \times 10}{7 \times 10} = \frac{10}{70}$$ $$\frac{1}{5} = \frac{1 \times 14}{5 \times 14} = \frac{14}{70}$$ $$\frac{1}{2} = \frac{1 \times 35}{2 \times 35} = \frac{35}{70}$$ 5. **Add the fractions:** $$\frac{10}{70} + \frac{14}{70} + \frac{35}{70} = \frac{10 + 14 + 35}{70} = \frac{59}{70}$$ 6. **Simplify the fraction:** 59 and 70 have no common factors other than 1, so $\frac{59}{70}$ is already in simplest form. **Final answer:** $$\boxed{\frac{59}{70}}$$