I can tell you without computation that for all , which pretty much settles it...
So, why? Because is the probability for a symmetric simple random walk on the integers to be back at its starting point at time (the binomial coefficient comes from choosing which steps go to the right); hence . Or just because is the number of subsets of with elements, which is less than , the number of subsets of .
By the way, Stirling estimate gives if I remember well, hence the ratio and root tests would be unconclusive.