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Math Help - Finding radius of convergence on a series

  1. #1
    iva
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    Finding radius of convergence on a series

    I hope this is the right forum for this, it is a differential equations I'm doing, with series now which I'm reviewing before getting into the calculus of it.

    I have to find the radius and interval of convergence for this series, and i'm using a ratio test but I can't get past a certain point. my original equation is:

    \sum_{n=1}^{\infty} \frac {(-1)^{n-1} x^{2n-1}} {(2n-1)!}

    I then applied the ratio test (i.e. n+1 series on top and original series on the bottom) and get this far:

    \lim_{n}^{\infty} \frac{-x}{2n} = \frac{-x}{2}\lim_{n}^{\infty} \frac{1}{n}

    Seeing as its the harmonic series whose sum is infinity i am baffled at how to simplify this so i can continue with my equation to get my radius and interval of conversion.

    Any advice really appreciated

    Thank you
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  2. #2
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    Re: Finding radius of convergence on a series

    Quote Originally Posted by iva View Post
    I hope this is the right forum for this, it is a differential equations I'm doing, with series now which I'm reviewing before getting into the calculus of it.

    I have to find the radius and interval of convergence for this series, and i'm using a ratio test but I can't get past a certain point. my original equation is:

    \sum_{n=1}^{\infty} \frac {(-1)^{n-1} x^{2n-1}} {(2n-1)!}

    I then applied the ratio test (i.e. n+1 series on top and original series on the bottom) and get this far:

    \lim_{n}^{\infty} \frac{-x}{2n} = \frac{-x}{2}\lim_{n}^{\infty} \frac{1}{n}

    Seeing as its the harmonic series whose sum is infinity i am baffled at how to simplify this so i can continue with my equation to get my radius and interval of conversion.

    Any advice really appreciated

    Thank you
    You have not applied the ratio test correctly.

    By the ratio test, the series will be convergent when \displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1.
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  3. #3
    Super Member girdav's Avatar
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    Re: Finding radius of convergence on a series

    The ratio test is a good idea. Fix x\neq 0 and put a_n:=\frac{(-1)^{n-1}x^{2n-1}}{(2n-1)!}. We have to compute \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|. We get 0 hence the radius of convergence is infinite.
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  4. #4
    iva
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    Re: Finding radius of convergence on a series

    Thanks guys, was my calculation of the ratio correct though ? because the limit for 1/n as n tends to infinity works out to 0. So I can conclude that there is no radius of convergence as the series is divergent?
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  5. #5
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    Re: Finding radius of convergence on a series

    You did not apply the ratio test correctly for several reasons. First, because the ratio test only applies to series of positive numbers. That is why girdav has the absolute value in \lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|. You should not have that "-".

    Also, you have done the cancelation wrong. |a_n|= \frac{x^{2n-1}}{(2n-1)!} but a_{n+1}= \frac{x^{2(n+1)-1}}{(2(n+1)-1)!}= \frac{x^{2n+1}}{(2n+1)!}. Now the ratio is
    \left|\frac{a_{n+1}}{a_n}\right|= \frac{x^{2n+1}}{(2n+1)!}\frac{(2n-1)!}{x^{2n-1}}= \frac{x^{2n+1}}{x^{2n-1}}\frac{(2n-1)!}{(2n+1)!}
    = x^{(2n+1)- (2n-1)}\frac{(2n-1)!}{(2n+1)(2n)(2n-1)!}= \frac{x^2}{(2n+1)(2n)}

    Of course, that still goes to 0 so the "radius of convergence" is infinite- this series converges for all x.

    In fact, it is just the MacLaurin series for cos(x).
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  6. #6
    iva
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    Re: Finding radius of convergence on a series

    Gosh, finally got it after almost posting another question and then figuring it out while doing so. Thanks so much!
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