Does the infinite geometric series converge or diverge? 1 + 3/4 + 9/16 + 27/64 + 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 by the previous term.Calculate Common Ratio: Looking at the series 1 + 3/4 + 9/16 + 27/64 + ..., we can find the common ratio by dividing the second term (3/4) by the first term (1), which gives us 3/4. We can check this by dividing the third term (9/16) by the second term (3/4) and so on.Check Absolute Value: The common ratio r = 3/4. Now, we need to check if the absolute value of r is less than 1 to determine if the series converges.Apply Convergence Test: Since the absolute value of 3/4 is less than 1 (|3/4| = 0.75 < 1), the infinite geometric series converges according to the convergence test for geometric series.Final Conclusion: Therefore, the correct choice is (A) converge.

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Does the infinite geometric series converge or diverge? $\newline$ $1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \ldots$ $\newline$ 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 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \ldots

\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 by the previous term.
Calculate Common Ratio: Looking at the series $1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \ldots$ , we can find the common ratio by dividing the second term $(\frac{3}{4})$ by the first term $(1)$ , which gives us $\frac{3}{4}$ . We can check this by dividing the third term $(\frac{9}{16})$ by the second term $(\frac{3}{4})$ and so on.
Check Absolute Value: The common ratio $r = \frac{3}{4}$ . Now, we need to check if the absolute value of $r$ is less than $1$ to determine if the series converges.
Apply Convergence Test: Since the absolute value of $\frac{3}{4}$ is less than $1$ (|\frac{3}{4}| = 0.75 < 1), the infinite geometric series converges according to the convergence test for geometric series.
Final Conclusion: Therefore, the correct choice is $(A)$ converge.