Answer
Converges
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.
We have: $a_n=\dfrac{1 \cdot 3 \cdot ..........(2n-1)}{(2n-1)!} $ and $a_{n+1}=\dfrac{1 \cdot 3 \cdot ..........(2n-1)}{(2n-1)!} \times \dfrac{(2n+1)}{2n(2n+1)}$
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{n+1}}{a_n}|=\lim\limits_{k \to \infty} |\dfrac{1}{2n}| \\=0 \lt 1$
So, we can conclude that the given series converges absolutely by the ratio test.