- #1

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- 12

## 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_{n}x^{n}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_{n}x^{n}.

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*? If you have another cluster point, say lim inf |a

_{n}|^{1/n}_{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?