Does the infinite geometric series converge or diverge? 1 + 3 + 9 + 27 + Choices: (A)converge (B)diverge

Find Common Ratio: To determine if an infinite geometric series converges or diverges, we need to find the common ratio (r) of the series. The common ratio is the factor by which each term is multiplied to get the next term. In this series, we can find the common ratio by dividing the second term by the first term, the third term by the second term, and so on. 3 ÷ 1 = 3 9 ÷ 3 = 3 27 ÷ 9 = 3 The common ratio (r) is 3.Calculate Sum Formula: Now that we have the common ratio, we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), where S is the sum of the series, a is the first term, and r is the common ratio. This formula only applies if the absolute value of r is less than 1. Since the common ratio r is 3, which is greater than 1, the absolute value of r is not less than 1.Determine Convergence: Because the absolute value of the common ratio is greater than 1, the series does not meet the criteria for convergence. Therefore, the infinite geometric series 1 + 3 + 9 + 27 + ... diverges.

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Does the infinite geometric series converge or diverge? $\newline$ $1 + 3 + 9 + 27 + \ldots$ $\newline$ Choices: $\newline$ (A) converge $\newline$ (B) diverge

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Precalculus

Convergent and divergent geometric series

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Q. Does the infinite geometric series converge or diverge?

\newline

1 + 3 + 9 + 27 + \ldots

\newline

Choices:

\newline

(A) converge

\newline

(B) diverge

Find Common Ratio: To determine if an infinite geometric series converges or diverges, we need to find the common ratio $r$ of the series. The common ratio is the factor by which each term is multiplied to get the next term. $\newline$ In this series, we can find the common ratio by dividing the second term by the first term, the third term by the second term, and so on. $\newline$ $3 \div 1 = 3$ $\newline$ $9 \div 3 = 3$ $\newline$ $27 \div 9 = 3$ $\newline$ The common ratio $r$ is $3$ .
Calculate Sum Formula: Now that we have the common ratio, we can use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{(1 - r)}$ , where $S$ is the sum of the series, $a$ is the first term, and $r$ is the common ratio. This formula only applies if the absolute value of $r$ is less than $1$ . Since the common ratio $r$ is $3$ , which is greater than $1$ , the absolute value of $r$ is not less than $1$ .
Determine Convergence: Because the absolute value of the common ratio is greater than $1$ , the series does not meet the criteria for convergence. Therefore, the infinite geometric series $1 + 3 + 9 + 27 + \ldots$ diverges.