Given the expression $\frac{4k^3 - 2k^2 + n}{n^4}$, it seems we need to analyze it in the context of a Riemann sum. Usually, Riemann sums are used to approximate the integral of a function over an interval. Let's break down the steps to better understand how we can proceed.

Step-by-Step Solution

1. Expression of Riemann Sum: A Riemann sum typically has the form: $\sum_{k=1}^{n} f(x_k) \Delta x$ where $f(x_k)$ is the function evaluated at a specific point in each subinterval, and $\Delta x$ is the width of each subinterval.

2. Identify the components:

   • $f(x_k)$ in this case seems to be related to $\frac{4k^3 - 2k^2 + n}{n^4}$.
   • The variable $k$ usually denotes the subinterval index.
   • $n$ represents the number of subintervals.

3. Normalize the expression: Since $f(x_k)$ is given in terms of $k$ and $n$, we need to express $k$ in terms of $x$. Generally, $k$ can be related to $x$ as $k = x \cdot n$.

4. Consider the limit as $n \to \infty$: The integral is the limit of the Riemann sum as $n$ approaches infinity: $\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta x$

5. Evaluate $f(x)$ in the context of Riemann sum: For $f(x) = \frac{4k^3 - 2k^2 + n}{n^4}$, replace $k$ with $x \cdot n$: $f(x) = \frac{4(xn)^3 - 2(xn)^2 + n}{n^4} = \frac{4x^3 n^3 - 2x^2 n^2 + n}{n^4} = \frac{4x^3 n^3}{n^4} - \frac{2x^2 n^2}{n^4} + \frac{n}{n^4}$

   Simplifying each term: $f(x) = \frac{4x^3}{n} - \frac{2x^2}{n^2} + \frac{1}{n^3}$

   As $n \to \infty$, the terms involving $\frac{1}{n}$, $\frac{1}{n^2}$, and $\frac{1}{n^3}$ approach zero. Therefore: $f(x) \approx 0 \text{ as } n \to \infty$

6. Determine the integral: Since $f(x) \approx 0$ as $n \to \infty$, the integral of $f(x)$ over any interval will also approach zero.

Thus, the integral (which the Riemann sum approximates) of $\frac{4k^3 - 2k^2 + n}{n^4}$ as $n \to \infty$ is:

$\boxed{0}$

Would you like more details or have any questions?

Here are some example questions you might have:

1. How do I solve quadratic equations?
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Tip: When working with limits, always check if the terms in your expression simplify to zero or infinity as the variable approaches the limit.