Problem 43 - College Algebra

Question

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 8(-0.3)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(\sum_{i=1}^{\infty} 8(-0.3)^{i-1}\) is approximately 6.15.

Step 1: Identify the first term (a) and the common ratio (r)

The first term (a) is the coefficient of the series which is 8. The common ratio (r) is the base of the exponent, in this case -0.3. So, we have a=8 and r=-0.3

Step 2: Check if series converges

The series will converge if -1< r< 1. In this case the common ratio r=-0.3 lies between -1 and 1, so the series converges and we can find the sum.

Step 3: Apply the sum formula

Applying the formula \(S = \frac{a}{1-r}\), with a=8 and r=-0.3, we get \(S = \frac{8}{1-(-0.3)}\)

Step 4: Simplify the equation

We have \(S = \frac{8}{1.3}\) after performing the subtraction in the denominator.

Step 5: Calculate the sum

Upon performing the division, we get \(S \approx 6.15\)