Please help to determine whether the series is convergent or divergent by Ratio Test

2^(n+1)/2(n+2)^2+1

When I found lim a(n+1)/a(n) = lim 2 (n is approaching infinity) =2

2 is greater than 1, series converges.

I probably made a mistake in lim calculations, if 2 is incorrect, will you please show me how to do it.

Thank you very much