Question

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Consider the infinite geometric series ∑∞n=1−4(13)n−1
. In this image, the lower limit of the summation notation is "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.

(3 points)

Answer 1

[AI-generated answer — Bot GPT 3.5]

a. The first four terms of the series are:

a1 = -4(13)^0 = -4(1) = -4
a2 = -4(13)^1 = -4(13) = -52
a3 = -4(13)^2 = -4(169) = -676
a4 = -4(13)^3 = -4(2197) = -8788

b. To determine if the series diverges or converges, we need to look at the common ratio r. In this case, r = -13. Since the absolute value of the common ratio is greater than 1, the series diverges.

c. If the series has a sum, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
Plugging in the values we have:
Sum = -4 / (1 - (-13))
Sum = -4 / (1 + 13)
Sum = -4 / 14
Sum = -2/7

Therefore, if the series had converged, the sum would have been -2/7.