Is it possible to know whether the following series converges without its formula? Given this series: 
$$\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{10} + \dfrac{1}{14} + \dfrac{1}{18} + \ldots$$
Is it possible to know if the series converges without its formula? I tried using the comparison test with both $1/n$ and $1/n^2$ but these don't yield results. The formula for this series is $1/[2(2n+1)]$, which I can use in other convergence tests, but if I didn't know that or couldn't find it, is it still possible to determine convergence without the formula?
Thanks

$$\dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{10} + \dfrac{1}{14} + \dfrac{1}{18} + \ldots > \\ 
\dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{12} + \dfrac{1}{16} + \dfrac{1}{20} + \ldots = \\
\dfrac{1}{4}\left(\dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \ldots \right)$$
