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!
Also, in the module I'm doing we've not been taught L'Hopital's rule, although I've come across it in further reading, and the lecturer said we will not be allowed to answer questions using it in the exam...