To solve the given limit, we need to interpret the sum [tex]\(\lim _{n \rightarrow \infty} \sum_{k=1}^n\left(4+\frac{3 k}{n}\right)^2\left(\frac{3}{n}\right)\)[/tex] as a Riemann sum for a definite integral. Here's the step-by-step process for that interpretation:
1. Identify the general form of a Riemann sum:
The Riemann sum for a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] typically looks like:
[tex]\[
\lim_{n \rightarrow \infty} \sum_{k=1}^n f(x_k^*) \Delta x
\][/tex]
where [tex]\( \Delta x = \frac{b - a}{n} \)[/tex], and [tex]\( x_k^* \)[/tex] represents sample points in each sub-interval.
2. Match the given sum to this form:
In our case, the given sum is:
[tex]\[
\sum_{k=1}^n \left(4 + \frac{3k}{n}\right)^2 \left(\frac{3}{n}\right)
\][/tex]
Here, [tex]\( \Delta x = \frac{3}{n} \)[/tex].
3. Determine the sample points [tex]\(x_k\)[/tex] and the interval:
Compare this sum with the form above:
- [tex]\( f(x_k^*) = \left(4 + \frac{3k}{n}\right)^2 \)[/tex]
- [tex]\( \Delta x = \frac{3}{n} \)[/tex]
Next, identify [tex]\( x_k \)[/tex]:
- Here, [tex]\( x = \frac{3k}{n} \)[/tex], let [tex]\( u = \frac{3k}{n} \)[/tex], which represents [tex]\( x \)[/tex] in the interval.
So, [tex]\( x = u \)[/tex], and the function [tex]\( f(x) \)[/tex] becomes:
[tex]\[
f(u) = \left(4 + u\right)^2
\][/tex]
4. Determine the limits of integration:
For the interval:
- The sum starts at [tex]\( k = 1 \)[/tex] and ends at [tex]\( k = n \)[/tex].
- When [tex]\( k = 1 \)[/tex], [tex]\( u = \frac{3}{n} \)[/tex] approaches [tex]\(0\)[/tex] as [tex]\( n \)[/tex] becomes very large, but since [tex]\(4 + 0 = 4\)[/tex], the lower limit becomes [tex]\(4\)[/tex].
- When [tex]\( k = n \)[/tex], [tex]\( u = \frac{3n}{n} = 3 \)[/tex] added to [tex]\(4\)[/tex], gives the upper limit [tex]\(7\)[/tex].
5. Write the integral form:
Using these limits and our function, the Riemann sum converts into the definite integral:
[tex]\[
\int_{4}^{7} (x+4)^2 \, dx
\][/tex]
Therefore, the integral that represents the limit is:
[tex]\[
\boxed{\int_{4}^{7} (x+4)^2 \, dx}
\][/tex]
