## 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

*n*th 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.