I'm really shaky with radius of convergence and limits when complex numbers are thrown into the mix, so I would be really grateful if someone could check I've answered these questions correctly! (I can't find any examples online that are quite as complicated)

1. Find the limit of (2n*e^{(( ln(n^2) + i*pi*n )/(( 16(n^2) + 5i ))^0.5)})/((4n^{2}+ 3in)^{(1/2)}) [From n=1 to infinity]

2. Compute the radius of convergence of the power series: (sum from n=1 to infinity) of a_{n}z^{n}, where a_{n}= (2n + 1)!(n + 2i)^{n}/(3n)!

My answers: 1. The limit does not exist. 2. R=27/4

Help would be veeeery much appreciated!