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Which of the following geometric series converges?
All three series converge, so the answer is D. The common ratios for each sequence are (I) -1/9, (II) -1/10, and (III) -1/3. Consider a geometric sequence with the first term a and common ratio |r| < 1. Then the n-th partial sum (the sum of the first n terms) of the sequence is
Multiply both sides by r :
Subtract the latter sum from the first, which eliminates all but the first and last terms:
Solve for :
Then as gets arbitrarily large, the term will converge to 0, leaving us with
So the given series converge to (I) -243/(1 + 1/9) = -2187/10 (II) -1.1/(1 + 1/10) = -1 (III) 27/(1 + 1/3) = 18
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