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Math Help - Radius of convergence of a series solution

  1. #1
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    Radius of convergence of a series solution

    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?
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  2. #2
    Senior Member vincisonfire's Avatar
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    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 <br /> <br />
y''+2y'+y=0<br />
,
    <br /> <br />
y_1=\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x<br />
is not a solution but
    <br /> <br />
y_1=\sum_{n=0}^{\infty}\frac{(-x)^n}{n!}<br />
is.
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  3. #3
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    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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