Does the infinite geometric series converge or diverge? 1 + 8 + 64 + 512 + 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 found by dividing any term in the series by the previous term. Let's take the second term (8) and divide it by the first term (1). r = 8 / 1 = 8Check Absolute Value: Now that we have the common ratio (r = 8), we can use the fact that an infinite geometric series converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). Since our common ratio is 8, which is greater than 1, the series diverges.Conclusion: Therefore, the correct choice is (B) diverge, because the common ratio is greater than 1.

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Does the infinite geometric series converge or diverge? $\newline$ $1 + 8 + 64 + 512 + \dots$ $\newline$ 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 + 8 + 64 + 512 + \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 found by dividing any term in the series by the previous term. $\newline$ Let's take the second term $8$ and divide it by the first term $1$ . $\newline$ $r = \frac{8}{1} = 8$
Check Absolute Value: Now that we have the common ratio $r = 8$ , we can use the fact that an infinite geometric series converges if and only if the absolute value of the common ratio is less than $1$ (|r| < 1). $\newline$ Since our common ratio is $8$ , which is greater than $1$ , the series diverges.
Conclusion: Therefore, the correct choice is (B) diverge, because the common ratio is greater than $1$ .