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Does the infinite geometric series converge or diverge?\newline1+75+4925+343125+…1 + \frac{7}{5} + \frac{49}{25} + \frac{343}{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+75+4925+343125+…1 + \frac{7}{5} + \frac{49}{25} + \frac{343}{125} + \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 the term before it.\newlineUsing the second term (75)(\frac{7}{5}) and the first term (1)(1), we calculate the common ratio (r)(r) as follows:\newliner=(75)1=75r = \frac{\left(\frac{7}{5}\right)}{1} = \frac{7}{5}
  2. Find absolute value: Find the absolute value of the common ratio. ∣r∣=∣75∣=75|r| = |\frac{7}{5}| = \frac{7}{5}
  3. Determine convergence: Determine if the given geometric series converges or diverges based on the absolute value of the common ratio.\newlineSince ∣r∣=75>1|r| = \frac{7}{5} > 1, the series diverges.

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