For what values of \( r \) does the sequence \( \left\{r^{n}\right\} \) converge? Diverge?
Select the correct choice below and fill in any answer boxes in your choice.
A. The sequence \( \left\{r^{n}\right\} \) converges for \( r \leq \square \), and diverges otherwise.
B. The sequence \( \left\{r^{n}\right\} \) converges for \( \square \leq r \leq \square \), and diverges otherwise.
OC. The sequence \( \left\{r^{n}\right\} \) converges for \( r>\square \), and diverges otherwise.
OD. The sequence \( \left\{r^{n}\right\} \) converges for \( \square
Ask by George Turner. in the United States
Mar 15,2025
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The sequence \( \{r^{n}\} \) converges if the absolute value of \( r \) is less than 1, specifically for \( -1 < r < 1 \). In this range, the terms get closer to zero as \( n \) increases. For \( r \leq -1 \) or \( r \geq 1 \), the sequence diverges, as the terms either oscillate without settling or grow infinitely. So the correct choice is B: The sequence \( \left\{r^{n}\right\} \) converges for \( -1 < r < 1 \), and diverges otherwise. Want to dive a bit deeper? The behavior of geometric sequences, like \( \left\{r^{n}\right\} \), is a classic example in calculus! Remember, determining convergence or divergence can set the stage for important concepts in analysis, such as limits and series summations. If you're curious about aspect related to infinite series, check out the concept of the Ratio Test or Root Test for series convergence. These tests can help you understand how to determine convergence properties for more complex functions and sequences that pop up in calculus!
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