The given expression asks for the limit of a sum as $n \to \infty$, and it is of the form:

$\lim_{n \to \infty} \sum_{i=1}^{n} \left( 7 + \frac{i}{n} \right) \left( \frac{3}{n} \right)$

This type of sum resembles a Riemann sum, which is often used to approximate integrals.

Step-by-Step Solution:

1. Interpret as a Riemann Sum

The sum can be written in a form suitable for interpretation as a Riemann sum: $S_n = \sum_{i=1}^{n} \left( 7 + \frac{i}{n} \right) \left( \frac{3}{n} \right)$ The term $\frac{3}{n}$ can be viewed as the width of the partition, and $\left( 7 + \frac{i}{n} \right)$ as the height of the function at each point.

The general form of a Riemann sum is: $\sum_{i=1}^{n} f(x_i) \Delta x$ where $\Delta x = \frac{b - a}{n}$ and $x_i = a + i \Delta x$.

2. Recognize the Integral

As $n \to \infty$, the sum becomes an integral. The expression inside the summation suggests that the function we are approximating is $f(x) = 7 + x$ and the partition is uniform over the interval $[0,1]$. Hence, we can write the integral as: $\int_0^1 (7 + x) \cdot 3 \, dx$

3. Compute the Integral

Now, let's compute the integral: $\int_0^1 (7 + x) \cdot 3 \, dx = 3 \int_0^1 (7 + x) \, dx$

Breaking this into two simpler integrals: $3 \int_0^1 7 \, dx + 3 \int_0^1 x \, dx$

• The first integral is: $3 \cdot 7 \int_0^1 1 \, dx = 3 \cdot 7 \cdot [x]_0^1 = 21 \cdot (1 - 0) = 21$

• The second integral is: $3 \int_0^1 x \, dx = 3 \cdot \left[ \frac{x^2}{2} \right]_0^1 = 3 \cdot \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = 3 \cdot \frac{1}{2} = \frac{3}{2}$

4. Final Answer

Adding both results together: $21 + \frac{3}{2} = \frac{42}{2} + \frac{3}{2} = \frac{45}{2}$

Thus, the limit of the sum as $n \to \infty$ is: $\boxed{\frac{45}{2}}$

Would you like more details or have any further questions?

Related Questions:

1. How do you compute a Riemann sum in general?
2. What are the conditions for converting a sum into an integral?
3. How would the sum change if the bounds of integration were different?
4. Can you apply this method to sums that involve quadratic terms?
5. What if the function inside the summation was more complex, like $7 + \frac{i^2}{n^2}$?

Tip:

When dealing with sums approaching infinity, always check if it can be interpreted as a Riemann sum, which converts the problem into an integral!