Use the ratio test to determine if $$ \sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4 n} n! $$ converges or diverges and justify your answer. Answer Attempt 1 out of 2 Using the ratio test, $$ \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\square \cdot \sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4 n} n! $$ converges $$ \sim $$ since the ratio is

Question

Use the ratio test to determine if $$ \sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4 n} n! $$ converges or diverges and justify your answer. Answer Attempt 1 out of 2 Using the ratio test, $$ \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\square \cdot \sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4 n} n! $$ converges $$ \sim $$ since the ratio is <1

UpStudy AI · Accepted Answer

We start by letting  
$$
a_n=\left(\frac{8}{3}\right)^{4n} n!
$$
and then apply the ratio test, which examines  
$$
L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.
$$

Step 1: Write out the ratio  
$$
\frac{a_{n+1}}{a_n}=\frac{\left(\frac{8}{3}\right)^{4(n+1)} (n+1)!}{\left(\frac{8}{3}\right)^{4n} n!}.
$$

Step 2: Simplify the expression  
Notice that  
$$
\left(\frac{8}{3}\right)^{4(n+1)}=\left(\frac{8}{3}\right)^{4n}\left(\frac{8}{3}\right)^4,
$$
and  
$$
\frac{(n+1)!}{n!}=n+1.
$$
Thus,  
$$
\frac{a_{n+1}}{a_n}=\left(\frac{8}{3}\right)^4 (n+1).
$$

Step 3: Compute the limit  
$$
L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty}\left(\frac{8}{3}\right)^4 (n+1).
$$
Since $$\left(\frac{8}{3}\right)^4$$ is a constant and $$\lim_{n\to\infty}(n+1)=\infty$$, we have  
$$
L=\infty.
$$

Step 4: Apply the ratio test conclusion  
The ratio test tells us that the series converges if $$L<1$$ and diverges if $$L>1$$. Here, since  
$$
L=\infty>1,
$$
the series diverges.

Thus, the series  
$$
\sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4n} n!
$$
diverges by the ratio test.
The series $$ \sum_{n=1}^{\infty}\left(\frac{8}{3}\right)^{4n} n! $$ diverges because the ratio test yields $$ L = \infty $$, which is greater than 1.

Use the ratio test to determine if converges or diverges
and justify your answer.
Answer Attempt 1 out of 2
Using the ratio test,
converges since the ratio is <1

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

Use the ratio test to determine if converges or diverges and justify your answer. Answer Attempt 1 out of 2 Using the ratio test, converges since the ratio is <1