• Support PF! Buy your school textbooks, materials and every day products Here!

Help me understand this theorem

  • Thread starter Jamin2112
  • Start date
  • #1

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 ∑anxn is given by

1/R = lim sup |an|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 = 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).
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?

Answers and Replies

  • #2
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...