Find all r for which the infinite geometric series \(2 + 6r + 18r^2 + 54r^3 + \dotsb.\) is defined. Enter all possible values of r, as an interval.

Since we have a geometric series, the possible values of r are in the interval (-1,1).

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Since each term is 3r times the preceding term, this will be a geometric series for whatever number you choose for r.

Interval form: (-infinity, infinity).

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Common ratio is 3r For the geometric series to converge, the absolute value of the common ratio has to be less than 1 |Common ratio| < 1 |3r| < 1 -1/3 < r < 1/3 or

(-1/3,1/3)