Calculus AB and Calculus BC

CHAPTER 10 Sequences and Series

Concepts and Skills

In this chapter, we review infinite series for BC Calculus students. Topics include

• tests for determining convergence or divergence,

• functions defined as power series,

• MacLaurin and Taylor series,

• and estimates of errors.

All of Chapter 10 is BC ONLY.

A. SEQUENCES OF REAL NUMBERS

A1. Definitions.

An infinite sequence is a function whose domain is the set of positive integers, and is often denoted simply by an. The sequence defined, for example, by Image is the set of numbers Image The elements in this set are called the terms of the sequence, and the nth or general term of this sequence is Image

A sequence an converges to a finite number L if Image

If an does not have a (finite) limit, we say the sequence is divergent.

EXAMPLE 1

Does the sequence Image converge or diverge?

SOLUTION: Image hence the sequence converges to 0.

EXAMPLE 2

Does the sequence Image converge or diverge?

SOLUTION: Image hence the sequence converges to Image

Topic will not be tested on the AP examination, but some understanding of the notation and terminology is helpful.

EXAMPLE 3

Does the sequence Image converge or diverge?

SOLUTION: Image hence the sequence converges to 1.

Note that the terms in the sequence Image are alternately smaller and larger than 1. We say this sequence converges to 1 by oscillation.

EXAMPLE 4

Does the sequence Image converge or diverge?

SOLUTION: Since Image the sequence diverges (to infinity).

EXAMPLE 5

Does the sequence an = sin n converge or diverge?

SOLUTION: Because Image sin n does not exist, the sequence diverges. However, note that it does not diverge to infinity.

EXAMPLE 6

Does the sequence an = (−1)n + 1 converge or diverge?

SOLUTION: Because Image does not exist, the sequence diverges.

Note that the sequence 1, −1, 1, −1,… diverges because it oscillates.