Answer
$$2/3\lt x\lt4/3$$
Work Step by Step
Given $$\sum_{n=0}^{\infty} 27^{n}(x-1)^{3 n+2}$$ Since $a_n =27^{n}(x-1)^{3 n+2}$ and $a_{n+1} =27^{n+1}(x-1)^{3 n+5}$, then \begin{aligned} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|\frac{27 (27)^{n}(x-1)^{3} (x-1)^{3 n+2}}{(27)^{n}(x-1)^{3 n+2}}\right|\\ &=\lim _{n \rightarrow \infty}\left|27 (x-1)^{3}\right|\\ & =\lim _{n \rightarrow \infty}\left|27 (x-1)^{3}\right|\\ &=27\left|(x-1)^{3}\right| \lim _{n \rightarrow \infty} 1 \\ &=27\left|(x-1)^{3}\right| \end{aligned} Then the series converges for $$27\left|(x-1)^{3}\right| \lt 1 \ \to \ 2/3\lt x\lt 4/3$$
Now, we check the end points.
For $x= 2/3 $ $$\sum_{n=0}^{\infty} 27^{n}(x-1)^{3 n+2} = \frac{1}{9}\sum_{n=1}^{\infty}(-1)^n $$
which is an alternating series and $\lim_{n\to \infty }|a_n| \neq 0 $ (divergent).
For $x=4/3 $ $$\sum_{n=0}^{\infty} 27^{n}(x-1)^{3 n+2} = \frac{1}{9}\sum_{n=1}^{\infty}1 $$
which diverges.
Hence, the interval of convergence is $$2/3\lt x\lt4/3$$
