Understanding R, c, and x in Power Series

[kdinser]

I'm not really having problems following the directions and arriving at the correct values, but I'm not really sure what those values mean or what I'm really figuring out.
R = radius of convergence
c = center of the series

Take this one for example.
$$\sum\frac{(2x)^{2n}}{(2n)!}$$

Subjecting this to the ratio test gives me:

$$\lim_{n\rightarrow \infty}\frac{(2x)^2}{(2n+1)(2n+2)} = 0$$

I get that it doesn't matter what we use for x, we will always get 0. Does that mean that R= all real numbers and is the domain of x?

I think I understand what it means when we get a finite value for R. Does it mean the series will converge if x is any value between -R and +R with c+R and c-R as the endpoints. But what does it mean when we plug in the endpoints values into the original power series and then test for convergence or divergence. Also, why is it useful for us to know this? How does this eventually get applied?

Thanks for any help.

 

[quantumdude]

I get that it doesn't matter what we use for x, we will always get 0. Does that mean that R= all real numbers and is the domain of x?

No, it means that R (the radius[/color] of convergence) is infinite. It is the interval[/color] of convergence that is all real numbers and is the domain of f(x) (not of x).

I think I understand what it means when we get a finite value for R. Does it mean the series will converge if x is any value between -R and +R with c+R and c-R as the endpoints.

Yes.

But what does it mean when we plug in the endpoints values into the original power series and then test for convergence or divergence.

You have to test for endpoint convergence because the ratio test doesn't tell you that bit of information. Remember that the ratio test is inconclusive when the ratio of the limit of |a_n+1/a_n|=1. Well, that equality just happens to correspond to the endpoints of the interval of convergence.

The ratio test tells you that the series converges absolutely inside the IOC, and it tells you that the series diverges outside of it. It just doesn't tell you what happens at the boundaries, so you have to test those independently.

Also, why is it useful for us to know this? How does this eventually get applied?

Among other places, it gets applied in the analysis of differential equations whose solutions are not elementary functions. In those cases, you get a power series solution, and it is necessary to know when such a solution is convergent.

 

[kdinser]

Thanks Tom, that and what I got out of class tonight clears things up nicely.

 

[karen03grae]

Fortunate

You are fortunate to understand these power series. I do not know how to think of them except to memorize the formulas. Then when they ask me a conceptual question, I am screwed.