In this section, you will learn how to determine whether a geometric series converges and diverges. You will also learn how to find the sum of convergent geometric series.

In this section, you will learn how to determine whether a geometric series converges and diverges. You will also learn how to find the sum of convergent geometric series.

Conclude by giving your students these challenges:

- Retiring to Paradise by NRICH
- What Numbers Can We Make Now? by NRICH
- Polygon Pictures by NRICH
- Time of Birth by NRICH
- Stair Climbing

This problem is harder, and thus should appear last. It requires the student to split the series in twain, which isn't something they're used to doing.

For the following convergent series, find the common ratio and the sum of the series.

\(\displaystyle \sum_{n = 0}^\infty \left(\dfrac{-1}{6}\right)^n\)

For each of the series below, determine whether the series converges. If it does, state the common ratio and the sum of the series. Otherwise, write "undefined."

\(\dfrac{1}{3} + \dfrac{2}{9} + \dfrac{1}{27} + \dfrac{2}{81} + \dfrac{1}{243} + \dfrac{2}{729} + \ldots\)

\(\displaystyle \sum_{n = 1}^\infty -2\left(\dfrac{5}{9}\right)^{n - 1}\)

\(\displaystyle \sum_{n = 0}^\infty 3\left(\dfrac{12}{13}\right)^n\)