1. **State the problem:** Arrange the given mathematical expressions from least to greatest. 2. **List the expressions:** - $\sum_{i=5}^7 i$ - $\frac{7\pi}{6}$ - $\int_{2}^{5} x \, dx$ - $\sqrt{12}$ - $e^{2}$ - $4!$ - $\frac{7}{11}$ - $\infty$ - $\log_{2}(13)$ 3. **Evaluate each expression:** - $\sum_{i=5}^7 i = 5 + 6 + 7 = 18$ - $\frac{7\pi}{6} \approx \frac{7 \times 3.1416}{6} = 3.665$ - $\int_{2}^{5} x \, dx = \left[ \frac{x^2}{2} \right]_2^5 = \frac{25}{2} - \frac{4}{2} = \frac{21}{2} = 10.5$ - $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \approx 3.464$ - $e^{2} \approx 7.389$ - $4! = 4 \times 3 \times 2 \times 1 = 24$ - $\frac{7}{11} \approx 0.636$ - $\infty$ is infinitely large - $\log_{2}(13)$: since $2^3=8$ and $2^4=16$, $\log_2(13)$ is between 3 and 4, approximately $3.7$ 4. **Order the values from least to greatest:** $$\frac{7}{11} \approx 0.636 < \sqrt{12} \approx 3.464 < \frac{7\pi}{6} \approx 3.665 < \log_2(13) \approx 3.7 < \int_2^5 x \, dx = 10.5 < \sum_{i=5}^7 i = 18 < e^2 \approx 7.389 < 4! = 24 < \infty$$ Note: $e^2$ is approximately 7.389, which is less than 18, so the correct order is: $$\frac{7}{11} < \sqrt{12} < \frac{7\pi}{6} < \log_2(13) < e^2 < \int_2^5 x \, dx < \sum_{i=5}^7 i < 4! < \infty$$ 5. **Final ordered list:** $$\frac{7}{11} < \sqrt{12} < \frac{7\pi}{6} < \log_2(13) < e^2 < 10.5 < 18 < 24 < \infty$$ where $10.5 = \int_2^5 x \, dx$, $18 = \sum_{i=5}^7 i$, and $24 = 4!$. This completes the ordering from least to greatest.