1. ## Power series convergence

Prove that f(x) = Σ (k=0 to ∞) a_k x^k has a positive radius of convergence.

Can someone give me some help on where to start with this?

Thanks

2. Originally Posted by taypez
Prove that f(x) = Σ (k=0 to ∞) a_k x^k has a positive radius of convergence.

Can someone give me some help on where to start with this?

Thanks
I really do not understand the question.

The radius of convergence R is a number such that |x|<R the series converges absolutely. If |x|>R the series diverges. Now we say R = 0 if no such real number exists such that |x|<R implies convergence. And we say R = +oo if no such real number exists such that |x|>R implies divergence.
So are you asking to prove that ever power series has: positive radius of convergence, zero radius of convergence, or infinite radius of convergence?

3. ## power series

Thanks. Unfortunately, the question was just as confusing to me. My thought was the same as yours that R had conditions and was not always positive. I thought maybe there was something that I was not understanding. I'm assuming now that I'm supposed to solve it for R>0. Who knows with this text!

Thanks again.

4. Prehaps we are given some facts about $\displaystyle a_n$?
That would give the question some meaning.
Is there more to the given?

5. That's the problem stated directly out of the book.

6. Originally Posted by taypez
That's the problem stated directly out of the book.
What textbook are you using?

Consider $\displaystyle \sum\limits_n {n!x^n }$ this has a radius of convergence $\displaystyle R=0$.
Here is a counter-example.