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Does the infinite geometric series converge or diverge?\newline1+23+49+827+…1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \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+23+49+827+…1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Find Common Ratio: To determine if the infinite geometric series converges or diverges, we need to find the common ratio rr of the series.\newlineThe common ratio is the factor by which each term is multiplied to get the next term.\newlineLooking at the series, we can see that each term is multiplied by 23\frac{2}{3} to get the next term.\newlineSo, the common ratio r=23r = \frac{2}{3}.
  2. Check Absolute Value: Now, we need to check if the absolute value of the common ratio is less than 11 for the series to converge.\newline∣23∣=23|\frac{2}{3}| = \frac{2}{3}, which is less than 11.\newlineSince the absolute value of the common ratio is less than 11, the series converges.
  3. Use Sum Formula: The formula for the sum of an infinite geometric series is S=a(1−r)S = \frac{a}{(1 - r)}, where SS is the sum, aa is the first term, and rr is the common ratio.\newlineSince the common ratio is less than 11, we can use this formula to confirm that the series converges to a finite sum.

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