Questions: Determine whether the geometric series is convergent or divergent. sum from n=1 to infinity of (10^n)/((-8)^(n-1)) If it is convergent, find its sum. (If the quantity diverges, enter Diverges.)

Determine whether the geometric series is convergent or divergent.
sum from n=1 to infinity of (10^n)/((-8)^(n-1))
If it is convergent, find its sum. (If the quantity diverges, enter Diverges.)

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

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 also: \[ r = \frac{10}{(-8)} = -1.25 \]

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.