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

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: \(\boxed{\text{The series } \sum_{n=1}^{\infty} \frac{n^{4}}{1.5^{n}} \text{ converges.}}\)