Thread: Power Series - lim sup proof

1. Power Series - lim sup proof

Let sum(a_nx^n) be a power series with a_n not zero, and assume L=lim|a_(n+1)/a_n| exists.
a) Show that if L is not zero, then the series converges for all x in (-1/L,1/L).
b) Show that if L=0, then the series onverges for all x in R
c) Show that a) and b) continue to hold if L is replaced by the limit
L'=lim(s_n) where s_n=sup{|a_(k+1)/a_k|:k>=n}
The value L' is called the limit superior or lim sup of the sequence |a_(n+1)/a_n|. It exists iff the sequence is bounded.
d) Show that if |a_(n+1)/a_n| is unbounded, then the original series converges only when x=0

I'm looking at this and I have no clue how to start.
Like for a), I start by assuming L is not zero.
So we have lim|a_(n+1)/a_n|is nonzero. Then I have trouble getting further.

2. I assume you know that this is the ratio test!
Ratio test - Wikipedia, the free encyclopedia

3. yeah I see that it's the ratio test, but don't see where the (-1/R,1/R) comes from and I guess I don't see where we use the a_nx^n

4. If you fix an $\displaystyle x\in\mathbb R$ and let $\displaystyle b_n :=a_nx^n$, what about $\displaystyle \displaystyle \lim_{n\to\infty}\left|\frac{b_{n+1}}{b_n}\right|$?

5. Then by ratio test, if L<1 series converges absolutely
If L>1 series does not converge

6. Originally Posted by mathematic
Then by ratio test, if L<1 series converges absolutely
If L>1 series does not converge
The condition is on $\displaystyle x$, not on $\displaystyle L$ because it is given by the sequence $\displaystyle \left\{a_n\right\}$.

7. I guess I don't understand then, because I thought a) was asking about L

8. Ok I figured out a) and b)
Now for c) I need to look at sup. I get that, but am unsure on dealing with. Like I know I use ratio test again.

9. and sup=smallest upper bound. It's just applying it that has me stuck