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Does the infinite geometric series converge or diverge?\newline1+4+16+64+…1 + 4 + 16 + 64 + \dots\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+4+16+64+…1 + 4 + 16 + 64 + \dots\newlineChoices:\newline(A) converge\newline(B) diverge
  1. Find Common Ratio: To determine if the infinite geometric series converges or diverges, we need to find the common ratio rr of the series.\newlineThe common ratio is found by dividing any term in the series by the preceding term.\newlineFor this series, we can use the second term 44 and the first term 11 to find the common ratio.\newliner=41=4r = \frac{4}{1} = 4
  2. Check Convergence Criterion: An infinite geometric series converges if the absolute value of the common ratio is less than 11 (∣r∣<1|r| < 1).\newlineSince we found that r=4r = 4, we can see that ∣r∣=∣4∣=4|r| = |4| = 4, which is not less than 11.\newlineTherefore, the series does not meet the convergence criterion.
  3. Series Diverges: Since the common ratio is greater than 11, the series diverges.\newlineThe terms of the series will continue to grow larger without bound, and the sum of the series cannot approach a finite number.

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