Radius of convergence of a series solution

**rebghb** · June 4th 2010, 04:30 AM

Hey everyone!

When finding a series solution for a certain ODE (at an ordinary point or a regular singular point), do we usually care about the radius of convergence or the lower bound of this radius?

Another thing. what's the lower bound of a solution at an ordinary point, say for the ODE $y''+2y'+y=0$ we know that $y_1=\sum_{n=0}^{\infty}\frac{x^n}{n!}$

Thanks?

**vincisonfire** · June 6th 2010, 06:58 AM

We would like to know the exact radius of convergence, but this is seldom possible. When you cannot find the radius of convergence, you try to find a lower bound on it. This will allow you to trust your solution in the given domain.
As for $ y''+2y'+y=0 $ ,
$ y_1=\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x $ is not a solution but
$ y_1=\sum_{n=0}^{\infty}\frac{(-x)^n}{n!} $ is.

**rebghb** · June 7th 2010, 07:22 AM

Yes, your absolutely right (I accidently typed in the exp(x) series expansion). But, then, what is the lower bound for the radius of convergence of the solution?? Does it even have a lwoer bound?
Best.

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