# Ch 12: L'Hopital's Rule, Integrals & Series in Calculus

### About This Chapter

## L'Hopital's Rule, Integrals & Series in Calculus - Chapter Summary

Watch videos in this chapter and find out why L'Hopital's Rule is kind of a big deal in certain areas of calculus. After dealing with L'Hopital, our instructors help you dust off your skills in working with various series. You'll learn about divergence and convergence and explore coefficients and polynomials. When viewing the lessons in this chapter, you will learn to:

- Understand L'Hopital's Rule and apply it in simple and complex cases
- Solve improper integrals
- Define partial sums and infinite series
- Calculate geometric series
- Work with power series and functions defined by power series
- Use Taylor series, polynomials, and coefficients

Each lesson in the chapter features a video with a multimedia lesson presented by our professional instructors. You can follow along with the video as humorous animation and real-world examples come together for an entertaining education experience. Use the lesson transcript to see the example problems worked out in print and to study the key terms and functions set in bold type. Take the quizzes and chapter exam to test your comprehension of the material and return to specific lessons when necessary. You don't even have to review the entire video the second time around: use the video tags in the timeline to jump directly to the specific problem you need to rework.

### 1. What is L'Hopital's Rule?

L'Hopital's Rule helps us solve limits that are not within the range of other problem-solving tools. Learn about the scenario when L'Hopital's Rule applies, explore the history of the rule, and understand how to use derivatives to solve these types of problems.

### 2. Applying L'Hopital's Rule in Simple Cases

Mathematicians use L'Hopital's rule to simplify the evaluation of limits. This lesson explores the three-step plan of L'Hopital's rule, providing multiple examples to aid in understanding.

### 3. Applying L'Hopital's Rule in Complex Cases

Mathematicians use L'Hopital's rule to simplify the evaluation of limits. This lesson explores the use of L'Hopital's rule in complex cases, providing multiple examples to aid in understanding.

### 4. Infinite Series & Partial Sums: Explanation, Examples & Types

An infinite series is the sum of terms in an infinitely long sequence, but taking the sum of terms in a finite portion of the sequence is called a partial sum. Explore these two concepts through examples of five types of series: arithmetic, geometric, harmonic, alternating harmonic, and telescopic.

### 5. How to Calculate a Geometric Series

A geometric series is found by combining the numbers found in the sequence, seen through a formula. See the proof behind this formula and how it can be solved, even when using an infinite series.

### 6. Power Series in X & the Interval of Convergence

A power series in 'x' involves factors where an 'X' is added to a constant, and raised to a power, forming infinite terms. Learn how to build a power series and explore how the summing of these terms and a ratio test identifies the interval of convergence.

### 7. Functions Defined by Power Series

A power series, an infinitely repeated pattern of a polynomial, has a center and convergence despite lacking an end-point. Explore the notation of these and see examples of convergence using the ratio test.

### 8. Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples

The Taylor series provides a method to solve a challenging math function. Explore the definition, equations, and examples of the Taylor series, including coefficients and polynomials. Learn about approximating a polynomial, as well as unsolvable integrals.

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### Other Chapters

Other chapters within the GRE Math: Study Guide & Test Prep course

- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards