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Does the infinite geometric series converge or diverge?\newline1+2+4+8+…1 + 2 + 4 + 8 + \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+2+4+8+…1 + 2 + 4 + 8 + \ldots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Identify common ratio: Identify the common ratio of the geometric series by comparing two consecutive terms.\newlineThe first term is 11 and the second term is 22.\newlineThe common ratio rr is obtained by dividing the second term by the first term: r=21=2r = \frac{2}{1} = 2.
  2. Determine absolute value: Determine the absolute value of the common ratio. ∣r∣=∣2∣=2|r| = |2| = 2
  3. Check convergence criteria: Check if the absolute value of the common ratio is less than 11 to determine if the series converges.\newlineSince ∣r∣=2>1|r| = 2 > 1, the series does not meet the convergence criterion for a geometric series.
  4. Conclude series status: Conclude whether the series converges or diverges based on the value of |r|\. Because \$|r| > 1, the infinite geometric series diverges.

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