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Does the infinite geometric series converge or diverge?\newline1+73+499+34327+…1 + \frac{7}{3} + \frac{49}{9} + \frac{343}{27} + \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+73+499+34327+…1 + \frac{7}{3} + \frac{49}{9} + \frac{343}{27} + \ldots\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 (73)(\frac{7}{3}) and divide it by the first term (1)(1):\newlineCommon Ratio (r)=731=73(r) = \frac{\frac{7}{3}}{1} = \frac{7}{3}
  2. Find Absolute Value: Find the absolute value of the common ratio.\newline|r| = |\frac{\(7\)}{\(3\)}|\(\newline|r| = \frac{77}{33}\newline)
  3. Determine Convergence: Determine if the given geometric series converges or diverges.\newlineA geometric series converges if the absolute value of the common ratio is less than 11.\newlineSince ∣r∣=73>1|r| = \frac{7}{3} > 1, the series diverges.

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