I'm having trouble understanding the justification for THEOREM XII in Taylor and Mann's book.
THEOREM XII. The radius of convergence R of a power series ∑anxn is given by
1/R = lim sup |an|1/n.
The Attempt at a Solution
Here is the proof which follows the theorem:
But why is it lim sup |an|1/n ???? If you have another cluster point, say lim inf |an|1/n, then that will give us a larger R, since it will make the denominator larger in R = 1 / (lim sup |an|1/n). When we look for the radius of convergence, we look for the largest R, right?Proof. We appeal to Cauchy's root test. Let un = anxn.
lim sup |un|1/n = |x| lim sup |an|1/n = |x| / R,
where R is defined by R = 1 / (lim sup |an|1/n).