Does Convergence at Zero Imply Global Convergence for Power Series?

[Hummingbird25]

Dear all

If a series e.g. a power series results in x convergering towards zero, can then one conclude that this series converge for all number if let's x belongs to R?

Sincerely Yours
Hummingbird25

 

[acceler8]

what do you mean by R? the ratio between terms?

 

[vsage]

I believe Hummingbird was referring to the set of real numbers (often denoted by R). The answer to the question (if I read it correctly, I had to read it a few times) is no also, look up "radius of convergence".

 

[HallsofIvy]

Dear all

If a series e.g. a power series results in x convergering towards zero, can then one conclude that this series converge for all number if let's x belongs to R?

Sincerely Yours
Hummingbird25

This makes no sense at all. "A power series results in x converging towards zero"? First of all, x does not "converge" toward anything. It is a variable. Second, I don't know what you mean by saying "a power series results" in that.

If I really had to guess, I would guess you are asking about the "ratio test". If, for any series of positive numbers
$$\Sum_{n=0}^\infnty a_n$$ the sequence $$\frac{a_{n+1}}{a_n}$$ converges to any number less than 1, then the series converges.

From that it follows that if, for the power series [tex]\Sum_{n=0}^\infty a_nx^n[/itex] and some specific x, the ratio $$\left|\frac{a_{n+1}}{a_n}\right|\left|x\right|$$ is less than 1 then the series converges for that x. In particular, if $$\left|\frac{a_{n+1}}{a_n}\right|$$ converges to 0 then the above will converge to 0 <1 for all x and so the power series converges for all x.