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 $$ r\square $$, and diverges otherwise. B. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ \square \leq r \leq \square $$, and diverges otherwise. D. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ r \leq \square $$, and diverges otherwise. E. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ \square

Question

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 $$ r>\square $$, and diverges otherwise. B. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ \square \leq r \leq \square $$, and diverges otherwise. D. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ r \leq \square $$, and diverges otherwise. E. The sequence $$ \left\{r^{n}\right\} $$ converges for $$ \square

UpStudy AI · Accepted Answer

We wish to determine for which real numbers $$ r $$ the sequence $$ \{r^n\}_{n=1}^{\infty} $$ converges and for which it diverges. **Step 1. Consider when $$|r| < 1$$.** If $$|r| < 1$$ then as $$ n \to \infty $$, the sequence satisfies $$ r^n \to 0. $$ Thus, the sequence converges (to 0). **Step 2. Consider when $$ r = 1$$.** In this case, every term in the sequence is $$ 1^n = 1. $$ So the sequence converges (to 1). **Step 3. Consider when $$ r = -1$$.** Then the sequence alternates: $$ -1, \, 1, \, -1, \, 1, \, \dots $$ Since the sequence does not approach a single number, it diverges. **Step 4. Consider when $$|r| > 1$$.** If $$|r| > 1$$ then as $$ n \to \infty $$, the terms $$ r^n $$ become arbitrarily large in absolute value (and if $$ r $$ is negative, they also alternate in sign). In either case, the sequence diverges. **Conclusion** The sequence converges if and only if $$ -1 < r \leq 1. $$ Among the provided choices, the one that corresponds to this is: E. The sequence $$ \{r^n\} $$ converges for $$ \boxed{-1} < r \leq \boxed{1} $$, and diverges otherwise. The sequence $$ \{r^n\} $$ converges for $$ -1 < r \leq 1 $$ and diverges otherwise.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions