Q. Find the sum of the infinite geometric series.−9−427−1681−64243+Write your answer as an integer or a fraction in simplest form.______
Identify Terms: First, we need to identify the first term a and the common ratio r of the geometric series.The first term is −9.To find the common ratio, we divide the second term by the first term: (−27/4)/(−9)=3/4.So, the common ratio r is 3/4.
Check Convergence: Next, we check if the absolute value of the common ratio is less than 1, which is a necessary condition for the convergence of an infinite geometric series.Since ∣43∣=0.75, which is less than 1, the series converges.
Use Sum Formula: Now, we can use the formula for the sum of an infinite geometric series, which is S=1−ra, where S is the sum, a is the first term, and r is the common ratio.Let's plug in the values: S=1−(43)−9.
Calculate Denominator: We calculate the denominator: 1−(43)=41.
Calculate Sum: Now, we calculate the sum: S=(−9)/(41).To divide by a fraction, we multiply by its reciprocal: S=(−9)×(14)=−36.
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