To evaluate the given limit by interpreting the sum as a Riemann sum, we will follow these steps:

Problem Statement:

$\lim_{n \to \infty} \frac{4}{n} \left( \frac{4}{n} + \frac{8}{n} + \frac{12}{n} + \dots + \frac{4n}{n} \right)$

Step 1: Rewrite the sum

The sum can be rewritten as: $\lim_{n \to \infty} \frac{4}{n} \sum_{k=1}^n \frac{4k}{n}.$

Step 2: Interpret as a Riemann sum

• The term $\frac{4}{n}$ represents the width of the subintervals.
• The term $\frac{4k}{n}$ represents the height of the rectangles at the $k$-th subinterval.

This sum approximates the integral of a function over the interval $[0, 4]$, where: $f(x) = 4x.$

Step 3: Set up the integral

The sum corresponds to the Riemann sum for the integral of $f(x) = 4x$ on $[0, 4]$: $\int_0^4 4x \, dx.$

Step 4: Evaluate the integral

Compute the integral: $\int_0^4 4x \, dx = \left[ 2x^2 \right]_0^4 = 2(4^2) - 2(0^2) = 2(16) - 0 = 32.$

Final Answer:

$\lim_{n \to \infty} \frac{4}{n} \sum_{k=1}^n \frac{4k}{n} = 32.$

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Would you like a detailed breakdown of any specific step? Here are some related questions:

1. What are the properties of Riemann sums that make them suitable for approximating integrals?
2. How does the choice of the function $f(x)$ influence the interpretation of the Riemann sum?
3. Can we generalize this method for sums involving powers of $k$?
4. What happens if the interval of integration is changed (e.g., to $[1, 5]$)?
5. How would this process differ if we used a midpoint or trapezoidal rule instead of left-hand endpoints?

Tip: Riemann sums are a bridge between discrete summation and continuous integration—mastering them helps in understanding the transition from finite sums to integrals.