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Does the infinite geometric series converge or diverge?\newline1+85+6425+512125+…1 + \frac{8}{5} + \frac{64}{25} + \frac{512}{125} + \dots\newlineChoices:\newline(A)converge\newline(B)diverge\newline

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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+85+6425+512125+…1 + \frac{8}{5} + \frac{64}{25} + \frac{512}{125} + \dots\newlineChoices:\newline(A)converge\newline(B)diverge\newline
  1. Identify Common Ratio: Identify the common ratio of the geometric series by dividing a term by its preceding term.\newlineTwo consecutive terms are 11 and 85\frac{8}{5}.\newline(85)/1=85(\frac{8}{5}) / 1 = \frac{8}{5}\newlineCommon Ratio (r):85(r): \frac{8}{5}
  2. Find Absolute Value: Find the absolute value of the common ratio.\newline|r| = |\frac{\(8\)}{\(5\)}|\(\newline|r| = \frac{88}{55}\newline
  3. Determine Convergence: Determine if the given geometric series converges or diverges.\newlineSince ∣r∣=85>1|r| = \frac{8}{5} > 1, the series diverges because the absolute value of the common ratio is greater than 11.

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