Basic Technical Mathematics, 11th Edition

19.3 Infinite Geometric Series

• Series • Infinity • Limit • Sum of an Infinite Geometric Series • Repeating Decimal

In the previous sections, we developed formulas for the sum of the first n terms of an arithmetic sequence and of a geometric sequence. The indicated sum of the terms of a sequence is called a series.

EXAMPLE 1 Illustrations of series

1. The indicated sum of the terms of the arithmetic sequence 2, 5, 8, 11, 14, … is the series $2\hspace{0.17em}+\hspace{0.17em}5\hspace{0.17em}+\hspace{0.17em}8\hspace{0.17em}+\hspace{0.17em}11\hspace{0.17em}+\hspace{0.17em}14\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\cdots \hspace{0.17em}$.

2. The indicated sum of terms of the geometric sequence $1\hspace{0.17em},\hspace{0.17em}{\displaystyle \frac{1}{2}}\hspace{0.17em},\hspace{0.17em}{\displaystyle \frac{1}{4}}\hspace{0.17em},\hspace{0.17em}{\displaystyle \frac{1}{8}}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em}$ is the series $1\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{1}{2}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{1}{4}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{1}{8}}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\cdots \hspace{0.17em}\text{.}$

The series associated with a finite sequence will sum up to a real number. The series associated with an infinite arithmetic sequence will not sum up to a real ...