**4.1 Sequences **

4.1.1 The Limit of a Sequence

4.1.2 Determining the Limit of a Sequence

4.1.3 The Squeeze and Absolute Value Theorems

**4.2 Monotonic and Bounded Sequences **

4.2.1 Monotonic and Bounded Sequences

**4.3 Infinite Series **

4.3.1 An Introduction to Infinite Series

4.3.2 The Summation of Infinite Series

4.3.3 Geometric Series

4.3.4 Telescoping Series

4.3.5 Applications of Series

**4.4 Convergence and Divergence **

4.4.1 Properties of Convergent Series

4.4.2 The nth-Term Test for Divergence

**4.5 The Integral Test **

4.5.1 An Introduction to the Integral Test

4.5.2 Examples of the Integral Test

4.5.3 Using the Integral Test

4.5.4 Defining p-Series

**4.6 The Direct Comparison Test **

4.6.1 An Introduction to the Direct Comparison Test

4.6.2 Using the Direct Comparison Test

**4.7 The Limit Comparison Test **

4.7.1 An Introduction to the Limit Comparison Test

4.7.2 Using the Limit Comparison Test

4.7.3 Inverting the Series in the Limit Comparison Test

**4.8 The Alternating Series **

4.8.1 Alternating Series

4.8.2 The Alternating Series Test

4.8.3 Estimating the Sum of an Alternating Series

**4.9 Absolute and Conditional Convergences **

4.9.1 Absolute and Conditional Convergence

**4.10 The Ratio and Root Tests **

4.10.1 The Ratio Test

4.10.2 Examples of the Ratio Test

4.10.3 The Root Test

**4.11 Polynomial Approximations of Elementary Functions **

4.11.1 Polynomial Approximation of Elementary Functions

4.11.2 Higher-Degree Approximations

**4.12 Taylor and Maclaurin Polynomials **

4.12.1 Taylor Polynomials

4.12.2 Maclaurin Polynomials

4.12.3 The Remainder of a Taylor Polynomial

4.12.4 Approximating the Value of a Function

**4.13 Taylor and Maclaurin Series **

4.13.1 Taylor Series

4.13.2 Examples of the Taylor and Maclaurin Series

4.13.3 New Taylor Series

4.13.4 The Convergence of Taylor Series

**4.14 Power Series **

4.14.1 The Definition of Power Series

4.14.2 The Interval and Radius of Convergence

4.14.3 Finding the Interval and Radius of Convergence: Part One

4.14.4 Finding the Interval and Radius of Convergence: Part Two

4.14.5 Finding the Interval and Radius of Convergence: Part Three

**4.15 Power Series Representations of Functions **

4.15.1 Differentiation and Integration of Power Series

4.15.2 Finding Power Series Representations by Differentiation

4.15.3 Finding Power Series Representations by Integration

4.15.4 Integrating Functions Using Power Series