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.

**Topics 10.1 – 10.2**

**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 tested 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, that 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 any one 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**

Type 10: Sequences and Series Questions

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series