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Does the infinite geometric series converge or diverge?\newline1+125+14425+1,728125+…1 + \frac{12}{5} + \frac{144}{25} + \frac{1,728}{125} + \dots\newlineChoices:\newline(A) converge\newline(B) diverge

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Convergent and divergent geometric seriesright chevron icon

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Q. Does the infinite geometric series converge or diverge?\newline1+125+14425+1,728125+…1 + \frac{12}{5} + \frac{144}{25} + \frac{1,728}{125} + \dots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Identify Common Ratio: Identify the common ratio of the geometric series by dividing a term by its preceding term.\newlineWe can take the second term (125)(\frac{12}{5}) and divide it by the first term (1)(1).\newline1251=125\frac{\frac{12}{5}}{1} = \frac{12}{5}
  2. Confirm Common Ratio: Now, let's confirm the common ratio by dividing the third term (144/25)(144/25) by the second term (12/5)(12/5).(144/25)/(12/5)=(144/25)×(5/12)=144/60=12/5(144/25) / (12/5) = (144/25) \times (5/12) = 144/60 = 12/5This confirms that the common ratio rr is indeed 12/512/5.
  3. Determine Convergence: Determine if the series converges or diverges based on the common ratio.\newlineSince the absolute value of the common ratio |r|\(\newline) is |\frac{\(12\)}{\(5\)}|\(\newline), which is greater than \(1\newline), the series diverges.

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