How to prove $ \int_{-\infty}^{+\infty}\left|x\right|^ke^{-(x-3)^2/2}dx $ is finite? How to prove this integral as following is finite?

$$
\int_{-\infty}^{+\infty}\left|x\right|^ke^{-(x-3)^2/2}dx
$$

k is a positive integer
Fix $n \in \mathbb{N}$, and consider $f_n (x)= \chi_{[-n,n]}(x) |x|^k \exp( - (x-3)^2 /2)$.
Then we have that $f_n$ increases pointwise to $f$ and thus we have by MCT
$$\lim_{n \to \infty} \int_{-\infty}^\infty f_n(x) \textrm{d}x = \int_{-\infty}^\infty \lim_{n \to \infty} f_n(x) \textrm{d}x = \int_{-\infty}^\infty f(x) \textrm{d}x.$$
Furthermore, we can find $\int_{-\infty}^\infty f_n(x) \textrm{d}x $ by using the power series of $\exp(- (x-3)^2 /2)$, and denote $g_{n,m}$ to be the partial sum till the $m^{\text{th}}$ term on $[-n,n]$ and $0$ outside $[-n,n]$.
As the $g_{n,m}$ converges pointwise to $f_n$ and the series $g_{n,m}$ is bounded by the integrable function $g_n(x) = \Lambda_n \chi_{[-n,n]}(x)$ where $\Lambda_n$ is a finite upper bound of $\{g_{n,m} ,f_{n}\}$, we have by DCT
$$ \lim_{m \to \infty} \int_{-\infty}^\infty g_{n,m}(x) \textrm{d}x = \int_{- \infty}^\infty f_n(x) \textrm{d}x.$$
And thus
$$ \int_{-\infty}^\infty f(x) \textrm{d}x = \lim_{n \to \infty} \lim_{m \to \infty} \int_{-n}^n g_{n,m}(x) \textrm{d}x.$$
You can use Taylor series representation for exponential function. Then interchange integration and summation with Fubinis theorem.