Question

determine whether the geometric series is convergent or divergent. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)
10 - 6 + 3.6 - 2.16 +...

Explanation:

Step1: Identify the first term and common ratio

For a geometric series \(a + ar+ar^{2}+\cdots\), in the series \(10 - 6+3.6 - 2.16+\cdots\), the first term \(a = 10\).
The common ratio \(r=\frac{-6}{10}=- 0.6\).

Step2: Check the convergence condition

A geometric series \(\sum_{n = 0}^{\infty}ar^{n}\) converges if \(|r|\lt1\).
Since \(|r|=|-0.6| = 0.6\lt1\), the series converges.

Step3: Use the sum formula for a geometric series

The sum formula for an infinite - geometric series is \(S=\frac{a}{1 - r}\) (when \(|r|\lt1\)).
Substitute \(a = 10\) and \(r=-0.6\) into the formula:
\(S=\frac{10}{1-(-0.6)}=\frac{10}{1 + 0.6}=\frac{10}{1.6}=\frac{100}{16}=\frac{25}{4}=6.25\)

Answer:

\(6.25\)