In Exercises 37–44, find the sum of each infinite geometric series. ∞ Σ (i = 1) 8(- 0.3)^(i -- 1)

Verified step by step guidance

1. Identify the first term of the series, \( a = 8 \).
2. Determine the common ratio \( r = -0.3 \).
3. Check if the series converges by ensuring \( |r| < 1 \).
4. Use the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \).
5. Substitute the values of \( a \) and \( r \) into the formula to find the sum.

Key Concepts

Infinite Geometric Series

An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely. It is defined by a first term 'a' and a common ratio 'r'. The series converges if the absolute value of the common ratio is less than one (|r| < 1), allowing us to calculate the sum using the formula S = a / (1 - r).

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. It is denoted as 'r' and is crucial for determining whether the series converges or diverges. If |r| < 1, the series converges, and we can find a finite sum; if |r| ≥ 1, the series diverges, meaning it does not have a finite sum.

Sum Formula for Infinite Series

The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula is applicable only when the series converges, which occurs when the absolute value of the common ratio is less than one. Understanding how to apply this formula is essential for solving problems involving infinite geometric series.