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Does the infinite geometric series converge or diverge?\newline1+83+649+51227+…1 + \frac{8}{3} + \frac{64}{9} + \frac{512}{27} + \dots\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+83+649+51227+…1 + \frac{8}{3} + \frac{64}{9} + \frac{512}{27} + \dots\newlineChoices:\newline(A)converge\newline(B)diverge
  1. Identify common ratio: Identify the common ratio of the geometric series by dividing a term by its preceding term.\newlineUsing the second term (83)(\frac{8}{3}) and the first term (1)(1), we calculate the common ratio (r)(r) as follows:\newliner=(83)/1=83r = (\frac{8}{3}) / 1 = \frac{8}{3}
  2. Calculate absolute value: Find the absolute value of the common ratio. ∣r∣=∣83∣|r| = |\frac{8}{3}| Since 83\frac{8}{3} is greater than 11, its absolute value is also greater than 11.
  3. Determine convergence or divergence: Determine if the given geometric series converges or diverges based on the absolute value of the common ratio.\newlineSince ∣r∣=83>1|r| = \frac{8}{3} > 1, the series diverges according to the convergence test for geometric series, which states that a geometric series converges if and only if ∣r∣<1|r| < 1.

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