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Does the infinite geometric series converge or diverge?\newline1+45+1625+64125+…1 + \frac{4}{5} + \frac{16}{25} + \frac{64}{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+45+1625+64125+…1 + \frac{4}{5} + \frac{16}{25} + \frac{64}{125} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Identify common ratio: Identify the common ratio of the geometric series by dividing two consecutive terms.\newlineThe second term is 45\frac{4}{5} and the first term is 11.\newlineCommon ratio (r)=451=45(r) = \frac{\frac{4}{5}}{1} = \frac{4}{5}
  2. Calculate absolute value: Calculate the absolute value of the common ratio.\newline|r| = |\frac{4}{5}|\(\newline|r| = \frac{4}{5}\)
  3. Determine convergence: Determine if the geometric series converges or diverges based on the absolute value of the common ratio.\newlineSince ∣r∣=45<1|r| = \frac{4}{5} < 1, the series converges according to the convergence test for geometric series.

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