To solve the problem, let’s analyze the given expression:

$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n - k}{n^2} \cos\left(\frac{4k}{n}\right)$

This expression resembles a Riemann sum. To convert it into an integral, we need to interpret the terms in the summation accordingly.

Step-by-Step Solution:

1. Rewrite the Sum as a Riemann Sum:

   • As $n \to \infty$, the sum can be interpreted as: \approx \int_{0}^{1} (1 - x) \cos(4x) \, dx$$
   • Here, we let $x = \frac{k}{n}$ so that $\Delta x = \frac{1}{n}$.
   • Then, $\frac{n - k}{n} = 1 - x$ and $\frac{4k}{n} = 4x$.

2. Set Up the Integral: $\int_{0}^{1} (1 - x) \cos(4x) \, dx$

3. Integrate by Parts: To solve this integral, use integration by parts where:

   • Let $u = 1 - x$ and $dv = \cos(4x) \, dx$.
   • Then $du = -dx$ and $v = \frac{\sin(4x)}{4}$.

   Using the integration by parts formula $\int u \, dv = uv - \int v \, du$: $\int_{0}^{1} (1 - x) \cos(4x) \, dx = \left[ (1 - x) \frac{\sin(4x)}{4} \right]_{0}^{1} + \int_{0}^{1} \frac{\sin(4x)}{4} \, dx$

4. Evaluate Each Part:

   • For the first term: $\left[ (1 - x) \frac{\sin(4x)}{4} \right]_{0}^{1} = (0) - \left(\frac{1 \cdot \sin(0)}{4}\right) = 0$
   • For the second term, integrate $\frac{\sin(4x)}{4}$ from $0$ to $1$: $\int_{0}^{1} \frac{\sin(4x)}{4} \, dx = \frac{1}{4} \int_{0}^{1} \sin(4x) \, dx = \frac{1}{4} \left[ -\frac{\cos(4x)}{4} \right]_{0}^{1}$ $= \frac{1}{4} \cdot \left( -\frac{\cos(4)}{4} + \frac{1}{4} \right) = \frac{1}{16} (1 - \cos(4))$

Final Answer:

Thus, the value of the limit is:

$\frac{1}{16} (1 - \cos(4))$

Correct Option:

The answer is (d) $\frac{1}{16} (1 - \cos 4)$.

Would you like further explanation or have any questions?

Related Questions:

1. How does converting a summation to a Riemann integral help in finding limits?
2. Can integration by parts always be used with trigonometric functions?
3. What are some typical forms that indicate a problem can be solved using a Riemann sum?
4. How do we know which substitution to make when turning a discrete sum into an integral?
5. What is the significance of the $\cos(4)$ term in this particular limit?

Tip:

When working with Riemann sums, identifying the correct form of $f(x)$ and $dx$ in the context of the integral is key to solving the problem correctly.