Real Analysis Question regarding Series

[christianrhiley]

Homework Statement

Let {a_n} be a monotonically decreasing sequence of positive real numbers with lim a_n = 0. Show the radius of convergence of $$\sum$$a_nx$$^{}n$$ is at least 1.

The Attempt at a Solution

I have no real attempt at a solution since I'm unsure how to proceed. I've tried using the power series definition and using a fixed x_0, but I get nowhere using this method. Can someone outline how this proof might look? Thanks.

 

[morphism]

The radius of convergence is given by
$$\lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|.$$

If {a_n} is decreasing and lim a_n = 0, then what can we say about this limit?

 

[e(ho0n3]

Is the fact that $a_n = 0$ as n approaches infinity needed? I find no reason for it.

 

[christianrhiley]

Solved It!

Solved It!