Does the infinite geometric series converge or diverge? 1 + 9 + 81 + 729 + 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 found by dividing any term in the series by the previous term. Let's find the common ratio using the first two terms: r = 9 / 1Calculate Value of r: Now, let's calculate the value of r: r = 9 / 1 = 9Check Absolute Value: An infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| 1), the series diverges.

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Does the infinite geometric series converge or diverge? $\newline$ $1 + 9 + 81 + 729 + \dots$ $\newline$ 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 + 9 + 81 + 729 + \dots

\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 found by dividing any term in the series by the previous term. $\newline$ Let's find the common ratio using the first two terms: $\newline$ $r = \frac{9}{1}$
Calculate Value of r: Now, let's calculate the value of r: $\newline$ $r = \frac{9}{1} = 9$
Check Absolute Value: An infinite geometric series converges if the absolute value of the common ratio is less than $1$ (|r| < 1). If $|r| \geq 1$ , the series diverges. $\newline$ Let's check the absolute value of our common ratio: $\newline$ $|r| = |9| = 9$
Series Diverges: Since the absolute value of the common ratio is greater than $1$ (9 > 1), the series diverges.