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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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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.

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\newline

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\newline

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