To write the left Riemann sum using sigma notation for the function $f(x) = e^x$ over the interval $[0, \ln 10]$ with $n = 30$, we proceed as follows:

Part 1: Sigma Notation

1. Interval Width: $\Delta x = \frac{\ln 10 - 0}{n} = \frac{\ln 10}{30}$

2. Left Endpoint: The left endpoint of each subinterval can be expressed as: $x_k = 0 + (k-1) \Delta x = (k-1) \frac{\ln 10}{30}$

3. Left Riemann Sum: Thus, the left Riemann sum in sigma notation is: $L_n = \sum_{k=1}^{30} f(x_k) \Delta x = \sum_{k=1}^{30} e^{(k-1) \frac{\ln 10}{30}} \cdot \frac{\ln 10}{30}$

   Therefore, the correct choice is: D. $\sum_{k=1}^{30} e^{(k-1) \frac{\ln 10}{30}} \cdot \frac{\ln 10}{30}$

Part 2: Evaluating the Left Riemann Sum

Now, let's evaluate this sum:

$L_n = \sum_{k=1}^{30} e^{(k-1) \frac{\ln 10}{30}} \cdot \frac{\ln 10}{30}$

Evaluation Steps

1. Calculate $\Delta x$: $\Delta x = \frac{\ln 10}{30} \approx \frac{2.302585}{30} \approx 0.0767528$

2. Calculate the sum: The term $e^{(k-1) \frac{\ln 10}{30}}$ simplifies to $10^{\frac{k-1}{30}}$.

3. Substitute into the sum: $L_n = \frac{\ln 10}{30} \sum_{k=1}^{30} 10^{\frac{k-1}{30}}$

4. Evaluate the geometric series: This is a geometric series with:

   • First term $a = 10^{0} = 1$
   • Common ratio $r = 10^{\frac{1}{30}}$
   • Number of terms $n = 30$

   The sum of the geometric series is given by: $S_n = a \frac{1 - r^n}{1 - r}$ Substituting the values: $S_{30} = 1 \cdot \frac{1 - (10^{\frac{1}{30}})^{30}}{1 - 10^{\frac{1}{30}}} = \frac{1 - 10}{1 - 10^{\frac{1}{30}}}$

   Therefore: $L_n = \frac{\ln 10}{30} \cdot \frac{1 - 10}{1 - 10^{\frac{1}{30}}} = \frac{\ln 10 \cdot (-9)}{30(1 - 10^{\frac{1}{30}})}$

Numerical Calculation

Now, you can use a calculator to evaluate this expression to find the left Riemann sum $L_n$.

After calculating the above expression, you should round your answer to three decimal places.

Summary

1. Sigma Notation: $\sum_{k=1}^{30} e^{(k-1) \frac{\ln 10}{30}} \cdot \frac{\ln 10}{30}$
2. Evaluate $L_n$ using the provided formula.

Would you like any specific calculations done or further explanations? Here are some related questions you might find interesting:

1. What is the interpretation of the Riemann sum in this context?
2. How does increasing $n$ affect the accuracy of the Riemann sum?
3. What is the difference between left and right Riemann sums?
4. How do you evaluate a Riemann sum for a different function?
5. What are the applications of Riemann sums in real-world scenarios?

Tip: Always check the convergence of your series by comparing it with an integral, especially for large $n$.