Does the infinite geometric series converge or diverge? 1 + 7 + 49 + 343 + 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. 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. Let's calculate the common ratio using the first two terms: r = 7 / 1 = 7Calculate Sum Formula: Now that we have the common ratio (r = 7), we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), where S is the sum, 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 (|r| 1), the infinite geometric series does not converge; instead, it diverges. The series will grow without bound as more terms are added.

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

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Math Problems

Precalculus

Convergent and divergent geometric series

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Q. Does the infinite geometric series converge or diverge?

\newline

1 + 7 + 49 + 343 + \dots

\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. $\newline$ 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. $\newline$ Let's calculate the common ratio using the first two terms: $\newline$ $r = \frac{7}{1} = 7$
Calculate Sum Formula: Now that we have the common ratio $r = 7$ , 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, $a$ is the first term, and $r$ is the common ratio. $\newline$ This formula only applies if the absolute value of $r$ is less than $1$ (|r| < 1). $\newline$ Since our common ratio $r$ is $7$ , 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 $1$ (|r| > 1), the infinite geometric series does not converge; instead, it diverges. $\newline$ The series will grow without bound as more terms are added.