### Summation and Integral EquivalenceGiven the summation formula:\[ L_n = \frac{4}{n} \sum_{i=1}^{n} \left[ 5 \left( 4 + (i-1) \frac{4}{n} \right)^{2} - 8 \left( 4 + (i-1) \frac{4}{n} \right)^{4} \right], \]we aim to express the limit as $ n \to \infty $ as a definite integral. Specifically, we need to determine the values of $ a $, $ b $, and $ f(x) $ in the following integral expression:\[ \int_{a}^{b} f(x) \, dx. \]### Explanation:1. **Interpretation of the Summation:** - Observe that the term $\frac{4}{n}$ in the expression can be related to the width of subintervals in a Riemann sum, which is commonly used in the definition of definite integrals. - The expression inside the summation resembles a function evaluated at discrete points, which are $4 + (i-1) \frac{4}{n}$.2. **Connecting to Definite Integral:** - As $ n \to \infty $, the term $\frac{4}{n}$ represents the differential element $dx$. - The variable $i$ runs from 1 to $n$, which translates the function evaluation across the entire interval for integration. 3. **Determining Limits of Integration $a$ and $b$:** - The points $4 + (i-1) \frac{4}{n}$ range from $4$ to $4 + (n-1) \frac{4}{n}$. - As $ n \to \infty $, the range approaches from $4$ to $8$. - Hence, the limits of integration are $a = 4$ and $b = 8$.4. **Determining the Function $f(x)$:** - The expression inside the summation can be interpreted as the function to be integrated. - The function being squared and raised to the fourth power is \(5\left(4 + (i-1) \frac{4}{

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Answered: Given Ln = 4 + express the limit as n →…

Given Ln = 4 + express the limit as n → ∞ as a definite integral, that is provide a, b and f(x) in the expression f(r)dr.

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section: Chapter Questions

Problem 22RE: Use summation notation to write the sum that results from adding the number 13 twenty times.

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Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

### Summation and Integral Equivalence

Given the summation formula:

\[ L_n = \frac{4}{n} \sum_{i=1}^{n} \left[ 5 \left( 4 + (i-1) \frac{4}{n} \right)^{2} - 8 \left( 4 + (i-1) \frac{4}{n} \right)^{4} \right], \]

we aim to express the limit as $ n \to \infty $ as a definite integral. Specifically, we need to determine the values of $ a $, $ b $, and $ f(x) $ in the following integral expression:

\[ \int_{a}^{b} f(x) \, dx. \]

### Explanation:

1. **Interpretation of the Summation:**
- Observe that the term $\frac{4}{n}$ in the expression can be related to the width of subintervals in a Riemann sum, which is commonly used in the definition of definite integrals.
- The expression inside the summation resembles a function evaluated at discrete points, which are $4 + (i-1) \frac{4}{n}$.

2. **Connecting to Definite Integral:**
- As $ n \to \infty $, the term $\frac{4}{n}$ represents the differential element $dx$.
- The variable $i$ runs from 1 to $n$, which translates the function evaluation across the entire interval for integration.

3. **Determining Limits of Integration $a$ and $b$:**
- The points $4 + (i-1) \frac{4}{n}$ range from $4$ to $4 + (n-1) \frac{4}{n}$.
- As $ n \to \infty $, the range approaches from $4$ to $8$.
- Hence, the limits of integration are $a = 4$ and $b = 8$.

4. **Determining the Function $f(x)$:**
- The expression inside the summation can be interpreted as the function to be integrated.
- The function being squared and raised to the fourth power is \(5\left(4 + (i-1) \frac{4}{

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