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

Identify common ratio: To determine if the infinite geometric series converges or diverges, we need to identify 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, each term is half the previous term, so the common ratio is 1/2.Check convergence criteria: We know that an infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1). Since the common ratio of our series is 1/2, and |1/2| < 1, the series converges.Apply 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. For this series, a = 1 and r = 1/2.Confirm convergence to 2: We can apply the formula to confirm that the series converges to a finite sum. S = 1 / (1 - 1/2) = 1 / (1/2) = 2. The series converges to the sum of 2.

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

\newline

Choices:

\newline

(A) converge

\newline

(B) diverge

Identify common ratio: To determine if the infinite geometric series converges or diverges, we need to identify 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. $\newline$ In this series, each term is half the previous term, so the common ratio is $\frac{1}{2}$ .
Check convergence criteria: We know that an infinite geometric series converges if the absolute value of the common ratio is less than $1$ (|r| < 1). $\newline$ Since the common ratio of our series is $\frac{1}{2}$ , and |\frac{1}{2}| < 1, the series converges.
Apply 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. $\newline$ For this series, $a = 1$ and $r = \frac{1}{2}$ .
Confirm convergence to $2$ : We can apply the formula to confirm that the series converges to a finite sum. $\newline$ $S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2.$ $\newline$ The series converges to the sum of $2$ .