Barrons AP Calculus - David Bock

EXAMPLE 4

Does the sequence converge or diverge?

SOLUTION: Since the sequence diverges (to infinity).

EXAMPLE 5

Does the sequence an = 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 an = (−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.

B. INFINITE SERIES

B1. Definitions.

Infinite series

If an is a sequence of real numbers, then an infinite series is an expression of the form

The elements in the sum are called terms; an is the nth or general term of the series.

EXAMPLE 7

A series of the form is called a p-series.

The p-series for p = 2 is

EXAMPLE 8

The p-series with p = 1 is called the harmonic series:

EXAMPLE 9

A geometric series has a first term, a, and common ratio of terms, r: