Chapter 1: Sequences and Series

Objectives:

1. Generalize, using inductive reasoning, a rule for determining the sum of an infinite geometric series.
   
2. Explain why an infinite geometric series is convergent or divergent.
   
3. Solve a problem that involves a geometric sequence or series.
   

An Infinite Geometric Series can either be divergent or convergent.

For a divergent series, the infinite sum does not approach a fixed value. (r \( \geq \) 1 or r \( \leq \) -1)

For a convergent series, the infinite sum does approach a finite value. ( -1 < r < 1)

The sum of an infinite series that converges can be shown to be:

S_{$\mathrm{\infty }$} = \( \frac{t_1}{1 - r} \), -1 < r < 1