1. **State the problem:** We need to order the following values from least to greatest: $$\sum_{i=5}^7 i, \sqrt{15}, \log_4(9), 7!, \infty, \frac{13}{17}, e^3, \int_2^9 x \, dx, \frac{2\pi}{3}$$ 2. **Calculate each value:** - Summation: $$\sum_{i=5}^7 i = 5 + 6 + 7 = 18$$ - Square root: $$\sqrt{15} \approx 3.873$$ - Logarithm: $$\log_4(9) = \frac{\ln 9}{\ln 4} \approx \frac{2.197}{1.386} \approx 1.585$$ - Factorial: $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$$ - Infinity: $$\infty$$ (largest) - Fraction: $$\frac{13}{17} \approx 0.7647$$ - Exponential: $$e^3 \approx 20.0855$$ - Definite integral: $$\int_2^9 x \, dx = \left[ \frac{x^2}{2} \right]_2^9 = \frac{9^2}{2} - \frac{2^2}{2} = \frac{81}{2} - 2 = 40.5 - 2 = 38.5$$ - Fraction with pi: $$\frac{2\pi}{3} \approx \frac{2 \times 3.1416}{3} = 2.0944$$ 3. **Order values from least to greatest:** $$\frac{13}{17} \approx 0.7647 < \log_4(9) \approx 1.585 < \frac{2\pi}{3} \approx 2.0944 < \sqrt{15} \approx 3.873 < 18 < e^3 \approx 20.0855 < 38.5 < 5040 < \infty$$ 4. **Final answer:** $$\frac{13}{17}, \log_4(9), \frac{2\pi}{3}, \sqrt{15}, \sum_{i=5}^7 i, e^3, \int_2^9 x \, dx, 7!, \infty$$