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

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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

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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