im learning about radius of convergence now but i still have yet to grasp its concept.

let "R hat" = (lim sup la_nl^1/n )^-1

assume that a_n is not 0 and the seq of positive numbers la_nl/la_n+1l converges to a limit. how do i prove that the lim la_nl/la_n+1l = 'R hat" where R hat is the radius of convergence of the series( sum from n = 1 to infinitiy a_n x^n?

thanks!