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Does the infinite geometric series converge or diverge?\newline1+7+49+343+…1 + 7 + 49 + 343 + \dots\newlineChoices:\newline(A) converge\newline(B) diverge

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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+7+49+343+…1 + 7 + 49 + 343 + \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 the factor by which each term is multiplied to get the next term.\newlineWe can find the common ratio by dividing the second term by the first term, the third term by the second term, and so on.\newlineLet's calculate the common ratio using the first two terms:\newliner=71=7r = \frac{7}{1} = 7
  2. Calculate Sum Formula: Now that we have the common ratio r=7r = 7, we can use the formula for the sum of an infinite geometric series, which is S=a1−rS = \frac{a}{1 - r}, where SS is the sum, aa is the first term, and rr is the common ratio.\newlineThis formula only applies if the absolute value of rr is less than 11 (∣r∣<1|r| < 1).\newlineSince our common ratio rr is 77, which is greater than 11, the absolute value of rr is not less than 11.
  3. Determine Convergence: Because the absolute value of the common ratio is greater than 11 (∣r∣>1|r| > 1), the infinite geometric series does not converge; instead, it diverges.\newlineThe series will grow without bound as more terms are added.

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