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Does the infinite geometric series converge or diverge?\newline1+8+64+512+…1 + 8 + 64 + 512 + \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+8+64+512+…1 + 8 + 64 + 512 + \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 previous term.\newlineLet's take the second term 88 and divide it by the first term 11.\newliner=81=8r = \frac{8}{1} = 8
  2. Check Absolute Value: Now that we have the common ratio r=8r = 8, we can use the fact that an infinite geometric series converges if and only if the absolute value of the common ratio is less than 11 (∣r∣<1|r| < 1).\newlineSince our common ratio is 88, which is greater than 11, the series diverges.
  3. Conclusion: Therefore, the correct choice is (B) diverge, because the common ratio is greater than 11.

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