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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} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge

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Convergent and divergent geometric seriesright chevron icon

Full solution

Q. Does the infinite geometric series converge or diverge?\newline1+34+916+2764+…1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Find Common Ratio: To determine if an infinite geometric series converges or diverges, we need to find the common ratio rr of the series. The common ratio is found by dividing any term by the previous term.
  2. Calculate Common Ratio: Looking at the series 1+34+916+2764+…1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \ldots, we can find the common ratio by dividing the second term (34)(\frac{3}{4}) by the first term (1)(1), which gives us 34\frac{3}{4}. We can check this by dividing the third term (916)(\frac{9}{16}) by the second term (34)(\frac{3}{4}) and so on.
  3. Check Absolute Value: The common ratio r=34r = \frac{3}{4}. Now, we need to check if the absolute value of rr is less than 11 to determine if the series converges.
  4. Apply Convergence Test: Since the absolute value of 34\frac{3}{4} is less than 11 (∣34∣=0.75<1|\frac{3}{4}| = 0.75 < 1), the infinite geometric series converges according to the convergence test for geometric series.
  5. Final Conclusion: Therefore, the correct choice is (A)(A) converge.

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