Find $M_{n}$ to three decimal places for the definite integral, using the indicated value of $n$. \[ \int_{2}^{4} \frac{2}{x} d x, n=8 \] \[ M_{n}= \]

Step 1 ：We are given a definite integral \(\int_{2}^{4} \frac{2}{x} dx\) and we are asked to find the Midpoint Rule approximation for this integral with n=8.

Step 2 ：The Midpoint Rule is a numerical method for approximating definite integrals. It works by dividing the area under the curve into rectangles and summing the areas of these rectangles. The height of each rectangle is determined by the value of the function at the midpoint of the base of the rectangle.

Step 3 ：The Midpoint Rule is given by the formula: \(M_{n} = \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_{i}}{2}\right)\) where \(\Delta x = \frac{b - a}{n}\) is the width of each rectangle, a and b are the limits of integration, and n is the number of rectangles.

Step 4 ：In this case, a = 2, b = 4, n = 8, and f(x) = \(\frac{2}{x}\).

Step 5 ：We calculate \(\Delta x = \frac{b - a}{n} = \frac{4 - 2}{8} = 0.25\).

Step 6 ：We then find the midpoints of the intervals: [2.125, 2.375, 2.625, 2.875, 3.125, 3.375, 3.625, 3.875].

Step 7 ：We substitute these midpoints into the function f(x) and sum the results, then multiply by \(\Delta x\).

Step 8 ：The Midpoint Rule approximation of the definite integral is \(\boxed{1.385}\).