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

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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

UpStudy AI · Accepted Answer

We analyze the sequence $$ \{r^n\} $$ as follows:

1. For $$ |r| < 1 $$:
   $$
   \lim_{n\to\infty} r^n = 0.
   $$
   So the sequence converges.

2. For $$ r = 1 $$:
   $$
   r^n = 1^n = 1 \quad \text{for all } n,
   $$
   hence the sequence converges (to 1).

3. For $$ r = -1 $$:
   $$
   r^n = (-1)^n,
   $$
   which alternates between 1 and -1. The sequence does not settle at a single number so it does not converge.

4. For $$ |r| > 1 $$:
   The terms $$ r^n $$ grow without bound (or oscillate with increasing magnitude when $$ r < -1 $$), so the sequence diverges.

Thus, the sequence converges when
$$
-1 < r \leq 1,
$$
and diverges otherwise.

This corresponds to choice D where the blanks are filled as:
$$
\text{The sequence } \{r^n\} \text{ converges for } -1 < r \leq 1, \text{ and diverges otherwise.}
$$
The sequence $$ \{r^n\} $$ converges for $$ -1 < r \leq 1 $$ and diverges otherwise.

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