We are asked to find values for $a$, $b$, and $f(x)$ such that the limit as $n \to \infty$ of the sum:

$\sum_{i=1}^{n} \frac{8i^6}{n^7}$

is equal to the integral of a function $f(x)$ over an interval $[a, b]$:

$\int_a^b f(x) \, dx.$

Step 1: Recognize the connection to Riemann sums

The given sum is reminiscent of a Riemann sum, which approximates the integral of a function. A typical Riemann sum for an integral of a function $f(x)$ over an interval $[a, b]$ is:

$\sum_{i=1}^{n} f\left(\frac{i}{n}\right) \cdot \frac{b - a}{n}.$

By comparing the structure of the given sum to the standard Riemann sum, we can infer that:

• The function inside the sum is $f\left(\frac{i}{n}\right)$.
• The term $\frac{8i^6}{n^7}$ corresponds to this, so $f(x)$ will be of a similar form.
• The $\frac{b-a}{n}$ term represents the width of the interval being partitioned.

Step 2: Rewrite the sum in Riemann sum form

To bring the sum into the form of a Riemann sum, let's factor out constants and express the sum in terms of $i/n$:

$\sum_{i=1}^{n} \frac{8i^6}{n^7} = \frac{8}{n} \sum_{i=1}^{n} \left(\frac{i}{n}\right)^6.$

This sum now has the form of a Riemann sum for the function $f(x) = 8x^6$, where $x = \frac{i}{n}$.

Step 3: Identify $a$, $b$, and $f(x)$

From the structure of the Riemann sum, we identify the following components:

• The function $f(x)$ being integrated is $f(x) = 8x^6$, since $\left(\frac{i}{n}\right)^6$ corresponds to $x^6$, and the factor of 8 is constant.
• The limits of integration $a$ and $b$ correspond to the values over which $x = \frac{i}{n}$ ranges. In this case, $i$ ranges from 1 to $n$, so