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Does the infinite geometric series converge or diverge?\newline1+32+94+278+…1 + \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \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+32+94+278+…1 + \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge\newline
  1. Identify Common Ratio: Identify the common ratio of the geometric series by dividing two consecutive terms.\newlineThe first term is 11 and the second term is 32\frac{3}{2}.\newlineCommon Ratio (r)(r) is calculated as 321=32\frac{\frac{3}{2}}{1} = \frac{3}{2}.
  2. Calculate Absolute Value: Determine the absolute value of the common ratio. ∣r∣=∣32∣|r| = |\frac{3}{2}| ∣r∣=32|r| = \frac{3}{2}
  3. Check Convergence: Check if the absolute value of the common ratio is less than 11 to determine if the series converges.\newlineSince ∣r∣=32>1|r| = \frac{3}{2} > 1, the series diverges.

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