Use the Root Test on ∑n=1∞(n2n+1)2n\sum_{n=1}^{\infty} \left(\frac{n}{2n+1}\right)^{2n}∑n=1∞(2n+1n)2n. What can you conclude?
Converges; ρ=14\rho = \frac{1}{4}ρ=41
Diverges; ρ=2\rho = 2ρ=2
Converges; ρ=12\rho = \frac{1}{2}ρ=21
Test is inconclusive; ρ=1\rho = 1ρ=1