Math 1b Topics, Spring 2000

Math 1b covers three main topics,

• Infinite Series
• Integration Techniques and Applications
• Differential Equations

Within each of these broad categories, there are many subcategories, and it may be useful for you to have a topical outline of the course. As we move through the course, we'll list here a rough outline of the subtopics that are being (or have been) covered.

For now, we have below the topic outline for Part I of the course (infinite series).

• Introduction to Infinite Series (7.4, 11.3)
  • Sigma notation
  • Formulas for some special sums
  • Basic concepts of infinite series
  • Geometric series
  • The Harmonic series
• Tests for Convergence (11.4, 11.6)
  • The divergence test
  • p-series
  • The comparison test
  • The ratio test
  • The root test
• Absolute and Conditional Convergence (11.7)
  • Alternating series definition and test for convergence
  • The concepts of absolute and conditional convergence
  • Ratio test for absolute convergence
• Power Series (11.8, 11.5)
  • Radius of convergence
  • Interval of convergence
  • Power series about a point other than zero (e.g., in (x-x₀) instead of x)
  • Functions defined by power series (e.g., Bessel functions)
  • Taylor and Maclaurin polynomials
  • Taylor and Maclaurin series
• Differentiation, Integration, and Manipulation of Power Series (11.10)
  • Differentiation
  • Integration
  • Using differentiation and integration of series to find Taylor series
  • Multiplication and division of power series
  • Modeling physical laws with series
• Convergence of Series and Numerical Derivations from Series (11.9)
  • Estimates for the remainder term for Taylor series
  • Using remainder term estimates to obtain numerical approximations (for exponentials, logarithms, etc.)
  • Binomial series