Find a, b and f(x) such that the limit as n aproaches infinity of (the sum of (8*i^6/n^7) between i=1 and n) is equal to the integral of f(x) between a and b

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

Find a, b and f(x) such that the limit as n approaches infinity of (the sum of (8*i^6/n^7) between i=1 and n) is equal to the integral of f(x) between a and b

Explore how to find the function f(x) and the limits a and b such that the limit as n approaches infinity of the sum (8*i^6/n^7) equals the integral of f(x) over the interval [a, b]. This problem covers key calculus concepts like Riemann sums, definite integrals, and limits, suitable for undergraduate students studying Calculus I/II.

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