Express The Limit As A Definite Integral.$\lim _{n \rightarrow \infty} \sum_{i=1}^n\left[\ln \left(1+\frac{3 I}{n}\right)-4\left(1+\frac{3 I}{n}\right)+2\right] \frac{3}{n} = $

Introduction

Limits and integrals are fundamental concepts in calculus, and they are closely related. In this article, we will explore how to express a limit as a definite integral. We will use the given problem as an example and work through the steps to find the solution.

The Problem

The problem asks us to express the limit as a definite integral:

$\lim {n \rightarrow \infty} \sum{i=1}^n\left[\ln \left(1+\frac{3 i}{n}\right)-4\left(1+\frac{3 i}{n}\right)+2\right] \frac{3}{n} = $

Understanding the Limit

To begin, let's understand the given limit. We have a sum of terms, each of which involves a natural logarithm and a linear function. The sum is multiplied by a constant factor of $\frac{3}{n}$.

As $n$ approaches infinity, the sum becomes a Riemann sum, which is a way of approximating a definite integral. The Riemann sum is a sum of areas of rectangles that approximate the area under a curve.

Expressing the Limit as a Definite Integral

To express the limit as a definite integral, we need to identify the function that is being integrated. In this case, the function is the natural logarithm function, which is given by:

$f(x) = \ln(x)$

The function is being evaluated at points of the form $1 + \frac{3i}{n}$, where $i$ ranges from $1$ to $n$. As $n$ approaches infinity, the points become a continuous interval from $1$ to $\infty$.

Finding the Definite Integral

To find the definite integral, we need to integrate the function $f(x) = \ln(x)$ with respect to $x$ over the interval from $1$ to $\infty$.

The definite integral is given by:

$\int_1^\infty \ln(x) dx = \lim_{b \to \infty} \int_1^b \ln(x) dx$

To evaluate the integral, we can use integration by parts. Let $u = \ln(x)$ and $dv = dx$. Then $du = \frac{1}{x} dx$ and $v = x$.

Using integration by parts, we get:

$\int \ln(x) dx = x \ln(x) - \int x \frac{1}{x} dx$

Evaluating the integral, we get:

$\int \ln(x) dx = x \ln(x) - x + C$

Now, we need to evaluate the definite integral:

$\int_1^\infty \ln(x) dx = \lim_{b \to \infty} \int_1^b \ln(x) dx$

Using the result from integration by parts, we get:

$\int_1^\infty \ln(x) dx = \lim_{b \to \infty} [x \ln(x) - x]_1^b$

Evaluating the limit, we get:

$\int_1^\infty \ln(x) dx = \lim_{b \to \infty} [b \ln(b) - b - \ln(1) + 1]$

Using L'Hopital's rule, we can evaluate the limit:

$\int_1^\infty \ln(x) dx = \lim_{b \to \infty} [b \ln(b) - b]$

Evaluating the limit, we get:

$\int_1^\infty \ln(x) dx = \infty$

Conclusion

In this article, we have seen how to express a limit as a definite integral. We used the given problem as an example and worked through the steps to find the solution.

The definite integral is given by:

$\int_1^\infty \ln(x) dx = \infty$

This result shows that the area under the curve of the natural logarithm function from $1$ to $\infty$ is infinite.

References

• [1] "Calculus" by Michael Spivak
• [2] "Real and Complex Analysis" by Walter Rudin
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Limits and Continuity" by James Stewart
• [2] "Integration" by George F. Simmons
• [3] "Calculus with Applications" by Margaret L. Lial

Glossary

• Riemann sum: A way of approximating a definite integral by summing the areas of rectangles that approximate the area under a curve.
• Definite integral: A mathematical expression that represents the area under a curve between two points.
• Integration by parts: A technique used to evaluate definite integrals by integrating one function and differentiating another function.
• L'Hopital's rule: A technique used to evaluate limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

FAQs

• Q: What is the difference between a limit and a definite integral? A: A limit is a value that a function approaches as the input gets arbitrarily close to a certain point. A definite integral is a mathematical expression that represents the area under a curve between two points.
• Q: How do I evaluate a definite integral? A: To evaluate a definite integral, you need to integrate the function with respect to the variable over the given interval.
• Q: What is the purpose of integration by parts? A: Integration by parts is a technique used to evaluate definite integrals by integrating one function and differentiating another function.
  Q&A: Expressing Limits as Definite Integrals =============================================

Q: What is the relationship between limits and definite integrals?

A: Limits and definite integrals are closely related. A limit is a value that a function approaches as the input gets arbitrarily close to a certain point. A definite integral is a mathematical expression that represents the area under a curve between two points. In many cases, a limit can be expressed as a definite integral.

Q: How do I know if a limit can be expressed as a definite integral?

A: To determine if a limit can be expressed as a definite integral, you need to check if the limit is a Riemann sum. A Riemann sum is a way of approximating a definite integral by summing the areas of rectangles that approximate the area under a curve.

Q: What is a Riemann sum?

A: A Riemann sum is a way of approximating a definite integral by summing the areas of rectangles that approximate the area under a curve. The Riemann sum is given by:

$\sum_{i=1}^n f(x_i) \Delta x$

where $f(x_i)$ is the function being integrated, $\Delta x$ is the width of each rectangle, and $n$ is the number of rectangles.

Q: How do I evaluate a Riemann sum?

A: To evaluate a Riemann sum, you need to sum the areas of the rectangles. The area of each rectangle is given by:

$A_i = f(x_i) \Delta x$

The sum of the areas of the rectangles is given by:

$\sum_{i=1}^n A_i = \sum_{i=1}^n f(x_i) \Delta x$

Q: What is the relationship between a Riemann sum and a definite integral?

A: A Riemann sum is an approximation of a definite integral. As the number of rectangles increases, the Riemann sum approaches the definite integral.

Q: How do I express a limit as a definite integral?

A: To express a limit as a definite integral, you need to identify the function being integrated and the interval over which the function is being integrated. You can then use the definition of a definite integral to express the limit as a definite integral.

Q: What are some common techniques for evaluating definite integrals?

A: Some common techniques for evaluating definite integrals include:

• Integration by parts
• Integration by substitution
• Integration by partial fractions
• Integration by trigonometric substitution

Q: What is integration by parts?

A: Integration by parts is a technique used to evaluate definite integrals by integrating one function and differentiating another function. The formula for integration by parts is given by:

$\int u dv = uv - \int v du$

Q: What is integration by substitution?

A: Integration by substitution is a technique used to evaluate definite integrals by substituting a new variable into the function being integrated. The formula for integration by substitution is given by:

$\int f(g(x)) g'(x) dx = \int f(u) du$

Q: What is integration by partial fractions?

A: Integration by partial fractions is a technique used to evaluate definite integrals by expressing a rational function as a sum of simpler fractions. The formula for integration by partial fractions is given by:

$\frac{f(x)}{g(x)} = \frac{A}{g_1(x)} + \frac{B}{g_2(x)} + \cdots$

Q: What is integration by trigonometric substitution?

A: Integration by trigonometric substitution is a technique used to evaluate definite integrals by substituting a trigonometric function into the function being integrated. The formula for integration by trigonometric substitution is given by:

$\int f(\sin(x)) \cos(x) dx = \int f(u) du$

Conclusion

In this article, we have seen how to express limits as definite integrals. We have also discussed some common techniques for evaluating definite integrals, including integration by parts, integration by substitution, integration by partial fractions, and integration by trigonometric substitution.

References

• [1] "Calculus" by Michael Spivak
• [2] "Real and Complex Analysis" by Walter Rudin
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Limits and Continuity" by James Stewart
• [2] "Integration" by George F. Simmons
• [3] "Calculus with Applications" by Margaret L. Lial

Glossary

• Riemann sum: A way of approximating a definite integral by summing the areas of rectangles that approximate the area under a curve.
• Definite integral: A mathematical expression that represents the area under a curve between two points.
• Integration by parts: A technique used to evaluate definite integrals by integrating one function and differentiating another function.
• Integration by substitution: A technique used to evaluate definite integrals by substituting a new variable into the function being integrated.
• Integration by partial fractions: A technique used to evaluate definite integrals by expressing a rational function as a sum of simpler fractions.
• Integration by trigonometric substitution: A technique used to evaluate definite integrals by substituting a trigonometric function into the function being integrated.

FAQs

• Q: What is the difference between a limit and a definite integral? A: A limit is a value that a function approaches as the input gets arbitrarily close to a certain point. A definite integral is a mathematical expression that represents the area under a curve between two points.
• Q: How do I evaluate a definite integral? A: To evaluate a definite integral, you need to integrate the function with respect to the variable over the given interval.
• Q: What is the purpose of integration by parts? A: Integration by parts is a technique used to evaluate definite integrals by integrating one function and differentiating another function.