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

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1. Identify the first term (a) of the infinite geometric series. Since the series is given by \( \sum_{i=1}^{\infty} 8(-0.3)^{i-1} \), the first term corresponds to \( i=1 \), so \( a = 8(-0.3)^{0} = 8 \).
2. Determine the common ratio (r) of the geometric series. The common ratio is the factor by which each term is multiplied to get the next term. Here, \( r = -0.3 \).
3. Check if the infinite geometric series converges. For convergence, the absolute value of the common ratio must be less than 1, i.e., \( |r| < 1 \). Since \( |-0.3| = 0.3 < 1 \), the series converges.
4. Use the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
5. Substitute the values of \( a = 8 \) and \( r = -0.3 \) into the formula to express the sum: \( S = \frac{8}{1 - (-0.3)} = \frac{8}{1 + 0.3} \).

Key Concepts

Infinite Geometric Series

An infinite geometric series is the sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. It converges only if the absolute value of the common ratio is less than 1, allowing the sum to approach a finite value.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to get the next term. It is crucial for determining whether the series converges and for calculating the sum of the series when |r| < 1.

Sum Formula for Infinite Geometric Series

The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by S = a / (1 - r). This formula allows us to find the total sum without adding infinitely many terms.