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Does the infinite geometric series converge or diverge?\newline1+94+8116+72964+…1 + \frac{9}{4} + \frac{81}{16} + \frac{729}{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+94+8116+72964+…1 + \frac{9}{4} + \frac{81}{16} + \frac{729}{64} + \ldots\newlineChoices:\newline(A)converge\newline(B)diverge
  1. Identify terms and ratio: Identify the first term aa and the common ratio rr of the geometric series. The first term is 11. To find the common ratio, divide the second term by the first term: (94)/1=94(\frac{9}{4}) / 1 = \frac{9}{4}.
  2. Calculate common ratio: Calculate the absolute value of the common ratio: ∣r∣=∣94∣=94|r| = \left|\frac{9}{4}\right| = \frac{9}{4}.
  3. Determine convergence: Determine if the series converges or diverges. A geometric series converges if the absolute value of the common ratio is less than 11. In this case, ∣r∣=94|r| = \frac{9}{4}, which is greater than 11.
  4. Series divergence: Since ∣r∣>1|r| > 1, the series diverges.

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