Answer
The domain of the given function is: $(-\infty, +\infty)$.
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$
1. For $l \lt 1$, the series is absolutely convergent.
2. For $l \gt 1$, the series is divergent.
3. For $l = 1$, the series is inconclusive.
Therefore, $l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|\\=\lim\limits_{k \to \infty} |\dfrac{1 \cdot 3 ........(2k+1)x^{k+1}}{(2k)!} \times \dfrac{(2k-2)!}{1 \cdot 3 .....(2k-1)x^k}| \\=0$
So, we can conclude that the given series converges absolutely if $l=0 \lt 1$. This means that the given series converges for all real numbers.
Therefore, the domain of the given function is: $(-\infty, +\infty)$.
