Unit 6: Infinite Series

Intro

  • A sum of infinitely many terms may converge to a finite value. This understanding will be further developed by exploring graphs, tables, and symbolic expressions for series that converge and diverge and for Taylor polynomials. It is paramount to understand previous concepts such as improper integrals.
  • The most important question related to this unit:
    • How can the sum of infinitely many discrete terms be a finite value or represent a continuous function?

Key Terms

Inquiry Activity

Podcast (coming soon)

6.1 Power Series

6.1.1 Convergent and Divergent Infinite Series

6.1.2 Geometric Series

6.1.3 Representing Functions as Power Series

6.2 Taylor Series and Taylor's Theorem

6.2.1 Finding Taylor Series for a Function

6.2.2 Finding Maclaurin Series for a Function

6.2.3 Finding Taylor Polynomial Approximations of Functions

6.2.4 Lagrange Error Bound

6.3 Radius of Convergence

6.3.1 The nth Term Test for Divergence

6.3.2 Ratio Test for Convergence

6.3.3 Radius and Interval of Convergence of Power Series

6.4 Testing Convergence at Endpoints

6.4.1 Integral Test for Convergence

6.4.2 Harmonic Series and p-Series

6.4.3 Comparison Tests for Convergence

6.4.4 Alternating Series Test for Convergence

6.4.5 Determining Absolute or Conditional Convergence

6.4.6 Alternating Series Error Bound