Problem 44
Question
Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 12(-0.7)^{i-1}$$
Step-by-Step Solution
Verified Answer
The sum of the infinite geometric series is approximately 7.06.
1Step 1: Identify the values
Identify the first term and the common ratio from the exercise. Here, the first term \( a = 12 \) and the common ratio \( r = -0.7 \).
2Step 2: Use the formula for sum of infinite geometric series
The formula for the sum of an infinite geometric series when the absolute value of the ratio is less than 1 is \( S = a / (1 - r) \). Substitute \( a = 12 \) and \( r = -0.7 \) into the formula.
3Step 3: Compute the sum
Plug in the values: \( S = 12 / (1 - (-0.7)) = 12 / 1.7 \). Simplify the fraction in order to represent the sum in the simplest terms.
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