Test the series below for convergence using the Ratio Test.
$$\sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{1.5}^n}$$
The limit of the ratio test simplifies to $\underset{n\to \mathrm{\infty }}{lim}|f(n)|$ where
$$f(n)=$$
The limit is:
(enter oo for infinity if needed)
Based on this, the series Select an answer $✓$

Step 1 ：The Ratio Test for convergence of a series states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit equals 1 or does not exist, the test is inconclusive.

Step 2 ：In this case, we need to find the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series. The nth term of the series is given by $f(n)=\frac{n^4}{{1.5}^n}$. The (n+1)th term of the series is given by $f(n+1)=\frac{(n+1{)}^4}{{1.5}^{n+1}}$.

Step 3 ：The ratio of the (n+1)th term to the nth term is given by $\frac{f(n+1)}{f(n)}=\frac{(n+1{)}^4/{1.5}^{n+1}}{n^4/{1.5}^n}=\frac{(n+1{)}^4}{n^4}\cdot \frac{{1.5}^n}{{1.5}^{n+1}}$.

Step 4 ：We need to find the limit as n approaches infinity of this ratio.

Step 5 ：The limit as n approaches infinity of the ratio of the (n+1)th term to the nth term of the series is 2/3, which is less than 1.

Step 6 ：Therefore, according to the Ratio Test, the series converges.

Step 7 ：Final Answer: $\boxed{{\displaystyle \text{The series }\sum _{n=1}^{\mathrm{\infty }}\frac{n^4}{{1.5}^n}\text{ converges.}}}$