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Does the infinite geometric series converge or diverge?\newline1+15+125+1125+…1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \ldots\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+15+125+1125+…1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Find Common Ratio: To determine if the infinite geometric series converges or diverges, we need to find the common ratio rr of the series.\newlineThe common ratio is the factor by which each term is multiplied to get the next term.\newlineIn this series, the second term is 15\frac{1}{5}, and the first term is 11. So, the common ratio rr is (15)/1=15\left(\frac{1}{5}\right) / 1 = \frac{1}{5}.
  2. Check Convergence Criteria: An infinite geometric series converges if the absolute value of the common ratio ∣r∣|r| is less than 11.\newlineIn this case, ∣r∣=∣15∣=15|r| = |\frac{1}{5}| = \frac{1}{5}, which is less than 11.
  3. Conclusion: Since the absolute value of the common ratio is less than 11, the series converges.\newlineTherefore, the correct choice is (A)(A) converge.

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