Does the infinite geometric series converge or diverge? 1 + 2/3 + 4/9 + 8/27 + Choices: (A)converge (B)diverge

Find Common Ratio: To determine if the 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. Looking at the series, we can see that each term is multiplied by 2/3 to get the next term. So, the common ratio r = 2/3.Check Absolute Value: Now, we need to check if the absolute value of the common ratio is less than 1 for the series to converge. |2/3| = 2/3, which is less than 1. Since the absolute value of the common ratio is less than 1, the series converges.Use Sum Formula: The formula for the sum of an infinite geometric series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Since the common ratio is less than 1, we can use this formula to confirm that the series converges to a finite sum.

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Does the infinite geometric series converge or diverge? $\newline$ $1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \ldots$ $\newline$ 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 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \ldots

\newline

Choices:

\newline

(A) converge

\newline

(B) diverge

Find Common Ratio: To determine if the infinite geometric series converges or diverges, we need to find the common ratio $r$ of the series. $\newline$ The common ratio is the factor by which each term is multiplied to get the next term. $\newline$ Looking at the series, we can see that each term is multiplied by $\frac{2}{3}$ to get the next term. $\newline$ So, the common ratio $r = \frac{2}{3}$ .
Check Absolute Value: Now, we need to check if the absolute value of the common ratio is less than $1$ for the series to converge. $\newline$ $|\frac{2}{3}| = \frac{2}{3}$ , which is less than $1$ . $\newline$ Since the absolute value of the common ratio is less than $1$ , the series converges.
Use Sum Formula: The formula for the sum of an infinite geometric series is $S = \frac{a}{(1 - r)}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ Since the common ratio is less than $1$ , we can use this formula to confirm that the series converges to a finite sum.