What is the Justification for Theorem XII in Taylor and Mann's Book?

[Jamin2112]

Homework Statement

I'm having trouble understanding the justification for THEOREM XII in Taylor and Mann's book.

Homework Equations

THEOREM XII. The radius of convergence R of a power series ∑a_nxⁿ is given by

1/R = lim sup |a_n|^1/n.​

​

The Attempt at a Solution

Here is the proof which follows the theorem:

Proof. We appeal to Cauchy's root test. Let un = a_nxⁿ.
Then

lim sup |u_n|^1/n = |x| lim sup |a_n|^1/n = |x| / R,​

where R is defined by R = 1 / (lim sup |a_n|^1/n).

But why is it lim sup |a_n|^1/n ? If you have another cluster point, say lim inf |a_n|^1/n, then that will give us a larger R, since it will make the denominator larger in R = 1 / (lim sup |a_n|^1/n). When we look for the radius of convergence, we look for the largest R, right?

 

[micromass]

You apply the Cauchy root test. This test will only work with limsup. To see why it doesn't work with liminf, you'll need to check the proof of the Cauchy root test...