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 a_n. The sequence defined, for example, by is the set of numbers The elements in this set are called the terms of the sequence, and the nth or general term of this sequence is

A sequence a_n converges to a finite number L if

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

EXAMPLE 1

Does the sequence converge or diverge?

SOLUTION: hence the sequence converges to 0.

EXAMPLE 2

Does the sequence converge or diverge?

SOLUTION: hence the sequence converges to

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

EXAMPLE 3

Does the sequence converge or diverge?

SOLUTION: hence the sequence converges to 1.

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

EXAMPLE 4

Does the sequence converge or diverge?

SOLUTION: Since the sequence diverges (to infinity).

EXAMPLE 5

Does the sequence a_n = sin n converge or diverge?

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

EXAMPLE 6

Does the sequence a_n = (−1)^{n + 1} converge or diverge?

SOLUTION: Because does not exist, the sequence diverges.

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