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Does the infinite geometric series converge or diverge?\newline1+35+925+27125+…1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge\newline

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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+35+925+27125+…1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge\newline
  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.\newlineLooking at the series, we can see that each term is multiplied by 35\frac{3}{5} to get the next term.\newlineSo, the common ratio r=35r = \frac{3}{5}.
  2. Apply Sum Formula: Now, we need to apply the formula for the sum of an infinite geometric series, which is S=a1−rS = \frac{a}{1 - r}, where SS is the sum of the series, aa is the first term, and rr is the common ratio.\newlineThe series converges if the absolute value of rr is less than 11 (∣r∣<1|r| < 1).\newlineIn this case, ∣r∣=∣35∣=35|r| = \left|\frac{3}{5}\right| = \frac{3}{5}, which is less than 11.
  3. Check Convergence: Since the absolute value of the common ratio is less than 11, the infinite geometric series converges.\newlineTherefore, the correct choice is (A)(A) converge.

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