Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{(1)(3) ....(2n-1)}{[(2)(4)....2n](3^n+1)}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(1)(3) ....(2n-1)(2n+1)}{[(2)(4)....2n(2n+2)](3^{n+1}+1)}}{\dfrac{(1)(3) ....(2n-1)}{[(2)(4)....2n](3^n+1)}}|$
Thus, we have $l= \lim\limits_{n \to \infty}|\dfrac{(2n+1)(3^n+1)}{(2n+2)(3^{n+1}+1)}|=\lim\limits_{n \to \infty}|\dfrac{(2+1/n)(1+1/3^n)}{(2+2/n)(3+1/3^n)}|$
So, $l=\dfrac{1}{3} \lt 1$
Hence, the series Converges by the ratio test.