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Does the infinite geometric series converge or diverge?\newline1+53+259+12527+…1 + \frac{5}{3} + \frac{25}{9} + \frac{125}{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+53+259+12527+…1 + \frac{5}{3} + \frac{25}{9} + \frac{125}{27} + \ldots\newlineChoices:\newline(A)converge\newline(B)diverge
  1. Find Common Ratio: To determine if the series converges or diverges, we need to find the common ratio rr of the geometric series. We can do this by dividing a term in the series by the term before it.\newlineLet's take the second term 53\frac{5}{3} and divide it by the first term 11:\newliner=53/1=53r = \frac{5}{3} / 1 = \frac{5}{3}
  2. Check Absolute Value: Now that we have the common ratio r=53r = \frac{5}{3}, we need to check its absolute value to determine if the series converges or diverges.\newlineThe absolute value of rr is ∣r∣=∣53∣=53|r| = \left|\frac{5}{3}\right| = \frac{5}{3}.
  3. Determine Convergence: For an infinite geometric series to converge, the absolute value of the common ratio must be less than 11. In this case, ∣r∣=53|r| = \frac{5}{3}, which is greater than 11.\newlineSince ∣r∣>1|r| > 1, the series diverges.

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