Question
Consider the infinite geometric series
n=1 infinity symbol -4(2/3)^n-1
a. Write the first four terms of the series.
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.
n=1 infinity symbol -4(2/3)^n-1
a. Write the first four terms of the series.
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.
Answers
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Answered by
GPT 3.5
a. The first four terms of the series are:
-4(2/3)^0 = -4
-4(2/3)^1 = -8/3
-4(2/3)^2 = -16/9
-4(2/3)^3 = -32/27
b. To determine if the series converges or diverges, we need to check if the common ratio (r) is between -1 and 1. In this case, the common ratio is 2/3 which is between -1 and 1. Therefore, the series converges.
c. The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio. Plugging in the values from the series, we have:
S = -4 / (1 - 2/3)
S = -4 / (1/3)
S = -12
Therefore, the sum of the infinite series is -12.
-4(2/3)^0 = -4
-4(2/3)^1 = -8/3
-4(2/3)^2 = -16/9
-4(2/3)^3 = -32/27
b. To determine if the series converges or diverges, we need to check if the common ratio (r) is between -1 and 1. In this case, the common ratio is 2/3 which is between -1 and 1. Therefore, the series converges.
c. The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio. Plugging in the values from the series, we have:
S = -4 / (1 - 2/3)
S = -4 / (1/3)
S = -12
Therefore, the sum of the infinite series is -12.
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