Question

determine whether the geometric series is convergent or divergent. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)
10 - 8 + 6.4 - 5.12 +...

Explanation:

Step1: Identify the first term \(a\) and common ratio \(r\)

For the geometric series \(10 - 8+6.4 - 5.12+\cdots\), the first term \(a = 10\).
To find the common ratio \(r\), we use the formula \(r=\frac{a_{n + 1}}{a_{n}}\). Taking \(n = 1\), \(a_{1}=10\) and \(a_{2}=- 8\), so \(r=\frac{-8}{10}=-0.8\)

Step2: Check the convergence condition

A geometric series \(\sum_{n = 0}^{\infty}a\times r^{n}\) converges if \(\vert r\vert<1\).
Since \(\vert r\vert=\vert-0.8\vert = 0.8<1\), 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 \(\vert r\vert<1\)).
Substitute \(a = 10\) and \(r=-0.8\) into the formula:

$$S=\frac{10}{1-(-0.8)}=\frac{10}{1 + 0.8}=\frac{10}{1.8}=\frac{100}{18}=\frac{50}{9}\approx5.56$$

Answer:

\(\frac{50}{9}\)