Questions: Determine whether the geometric series is convergent or divergent. ∑(n=1)^(∞) 10^n / (-8)^(n-1) convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Determine whether the geometric series is convergent or divergent.

∑(n=1)^(∞) 10^n / (-8)^(n-1)

convergent divergent

If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Transcript text: Determine whether the geometric series is convergent or divergent. \[ \sum_{n=1}^{\infty} \frac{10^{n}}{(-8)^{n-1}} \] convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.) $\square$

Solution

Solution Steps

To determine whether the given geometric series is convergent or divergent, we need to identify the common ratio $ r $ of the series. A geometric series converges if the absolute value of the common ratio $ |r| < 1 $. If the series converges, we can find its sum using the formula for the sum of an infinite geometric series: $ S = \frac{a}{1 - r} $, where $ a $ is the first term of the series.

Step 1: Identify the First Term and Common Ratio

The first term of the series is given by: \[ a = \frac{10}{(-8)} = -1.25 \] The common ratio is: \[ r = \frac{10}{(-8)} = -1.25 \]

Step 2: Determine Convergence or Divergence

To determine if the series converges, we check the absolute value of the common ratio: \[ |r| = |-1.25| = 1.25 \] Since $ |r| > 1 $, the series is divergent.