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Does the infinite geometric series converge or diverge?\newline1+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[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+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[B]diverge}
  1. Find common ratio: Find the common ratio rr by dividing the second term by the first term.34/1=34\frac{3}{4} / 1 = \frac{3}{4} Common ratio rr: 34\frac{3}{4}
  2. Check absolute value: Check if the absolute value of rr is less than 11.\newline∣r∣=∣34∣=34|r| = |\frac{3}{4}| = \frac{3}{4}\newlineSince 34\frac{3}{4} is less than 11, the series might converge.
  3. Apply convergence test: Apply the geometric series convergence test.\newlineIf ∣r∣<1|r| < 1, then the series converges.\newline∣r∣=34<1|r| = \frac{3}{4} < 1, so the series converges.

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