Unit 10 covers sequences and series. These are BC only topics (CED – 2019 p. 177 – 197). These topics account for about 17 – 18% of questions on the BC exam.

**Topic 10.1: Defining Convergent and Divergent Series.**

**Topic 10. 2: Working with Geometric Series. **Including the formula for the sum of a convergent geometric series.

**Topics 10.3 – 10.9 Convergence Tests**

The tests listed below are assessed on the BC Calculus exam. Other methods are not tested. However, teachers may include additional methods.

**Topic 10.3: The n^{th} Term Test for Divergence.**

**Topic 10.4: Integral Test for Convergence. **See Good Question 14

**Topic 10.5: Harmonic Series and p-Series. **Harmonic series and alternating harmonic series,

*p*-series.

**Topic 10.6: Comparison Tests for Convergence. **Comparison test and the Limit Comparison Test

**Topic 10.7: Alternating Series Test for Convergence.**

**Topic 10.8: Ratio Test for Convergence.**

**Topic 10.9: Determining Absolute and Conditional Convergence.** Absolute convergence implies conditional convergence.

**Topics 10.10 – 10.12 Taylor Series and Error Bounds**

**Topic 10.10: Alternating Series Error Bound.**

**Topic 10.11: Finding Taylor Polynomial Approximations of a Function.**

**Topic 10.12: Lagrange Error Bound.**

**Topics 10.13 – 10.15 Power Series**

**Topic 10.13: Radius and Interval of Convergence of a Power Series.** The Ratio Test is used almost exclusively to find the radius of convergence. Term-by-term differentiation and integration of a power series gives a series with the same center and radius of convergence. The interval may be different at the endpoints.

**Topic 10.14: Finding the Taylor and Maclaurin Series of a Function. **Students should memorize the Maclaurin series for , sin(*x*), cos(*x*), and *e ^{x}*.

**Topic 10.15: Representing Functions as Power Series. **Finding the power series of a function by differentiation, integration, algebraic processes, substitution, or properties of geometric series.

**Timing**

The suggested time for Unit 9 is about 17 – 18 BC classes of 40 – 50-minutes, this includes time for testing etc.

**Previous posts on these topics:**

**Before sequences**

Amortization Using finite series to find your mortgage payment. (Suitable for pre-calculus as well as calculus)

A Lesson on Sequences. An investigation, which could be used as early as Algebra 1, showing how irrational numbers are the limit of a sequence of approximations. Also, an introduction to the *Completeness Axiom. *

**Convergence Tests**

Which Convergence Test Should I Use? Part 1: Pretty much anyone you want!

Which Convergence Test Should I Use? Part 2: Specific hints and a discussion of the usefulness of absolute convergence

Good Question 14 on the Integral Test

**Sequences and Series**

Graphing Taylor Polynomials. Graphing calculator hints

New Series from Old 1: Substitution (Be sure to look at example 3)

New Series from Old 2: Differentiation

New Series from Old 3: Series for rational functions using long division and geometric series

Geometric Series – Far Out: An instructive “mistake.”

A Curiosity: An unusual Maclaurin Series

Synthetic Summer Fun Synthetic division and calculus including finding the (finite)Taylor series of a polynomial.

**Error Bounds**

Error Bounds: Error bounds in general and the alternating Series error bound, and the Lagrange error bound

The Lagrange Highway: The Lagrange error bound.

What’s the “Best” Error Bound?

**Review Notes**