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Math Help - radii of convergence question

  1. #1
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    radii of convergence question

    Question:

    A)
    s(x) = ∑ [x^(2n+1)] / [(2n+1)!]

    and c(x) = ∑(x^2n) / (2n!)

    Find the radii of convergence of these series and show that s'(x) =c(x) and c'(x) = s(x) for all xE R

    B)
    For every α E R, let hα (x) denote the function:
    s(x+α)c(α-x) + s(α-x)c(x+α)
    show that hα(x) has constant value s(2α)

    C)
    Use this result or by another means to show that for all β,γ E R

    s(β+γ)=s(β)c(γ)+s(γ)c(β)


    Solution:

    It is difficult for me to write it out on here because I don't know how to write all the maths properly like everybody else on maths forum because I am new!

    but for A) I think I must be going wrong because for both s(x) and c(x) the ratio test is telling me they are both null series and thus they converge for all x. Is this correct?

    b) I am unsure how to begin this part, maybe i need to use part a)

    any advice would be good as I'm having trouble even really starting this question!

    Thank you.
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    Quote Originally Posted by emjones View Post
    Question:

    A)
    s(x) = \sum\frac{x^{2n+1}}{(2n+1)!}

    and c(x) = \sum\frac{x^{2n}}{2n!}

    Find the radii of convergence of these series and show that s'(x) =c(x) and c'(x) = s(x) for all x\in\mathbb{R}

    B)
    For every \alpha\in\mathbb{R}, let h\alpha(x) denote the function:
    s(x+\alpha)c(\alpha-x) + s(\alpha-x)c(x+\alpha)
    show that h\alpha(x) has constant value s(2\alpha)

    C)
    Use this result or by another means to show that for all \beta,\gamma\in\mathbb{R}

    s(\beta+\gamma)=s(\beta)c(\gamma)+s(\gamma)c(\beta  )


    Solution:

    It is difficult for me to write it out on here because I don't know how to write all the maths properly like everybody else on maths forum because I am new!

    but for A) I think I must be going wrong because for both s(x) and c(x) the ratio test is telling me they are both null series and thus they converge for all x. Is this correct?

    b) I am unsure how to begin this part, maybe i need to use part a)

    any advice would be good as I'm having trouble even really starting this question!

    Thank you.
    Regarding (A), you are correct to note that both series converge for all x\in\mathbb{R}. That s(x)=c'(x) and s'(x)=c(x) follows directly from taking the derivative of each.

    In order to evaluate s(x+\alpha)c(\alpha-x) + s(\alpha-x)c(x+\alpha), you probably are supposed to simplify

    \sum\frac{(x+\alpha)^{2n+1}}{(2n+1)!}\sum\frac{(\a  lpha-x)^{2n}}{2n!}+\sum\frac{(\alpha-x)^{2n+1}}{(2n+1)!}\sum\frac{(x+\alpha)^{2n}}{2n!}.

    However, that would be a mess unless there is some kind of trick to doing it. I simply recognize that

    s(x)=\frac{e^x-e^{-x}}{2} and c(x)=\frac{e^x+e^{-x}}{2}, which means

    s(x+\alpha)c(\alpha-x) + s(\alpha-x)c(x+\alpha)

    =\frac{(e^{x+\alpha}-e^{-(x+\alpha)})(e^{\alpha-x}+e^{x-\alpha})+(e^{\alpha-x}-e^{x-\alpha})(e^{x+\alpha}+e^{-(x+\alpha)})}{4}

    =\frac{e^{2\alpha}-e^{-2\alpha}}{2}=s(2\alpha).

    Similar methods should work on the remainder of the problem. However, I'd be curious to know how to do this without actually recognizing the series.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Berkeley, California
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    Quote Originally Posted by emjones View Post
    Question:

    A)
    s(x) = ∑ [x^(2n+1)] / [(2n+1)!]

    and c(x) = ∑(x^2n) / (2n!)

    Find the radii of convergence of these series and show that s'(x) =c(x) and c'(x) = s(x) for all xE R

    B)
    For every α E R, let hα (x) denote the function:
    s(x+α)c(α-x) + s(α-x)c(x+α)
    show that hα(x) has constant value s(2α)

    C)
    Use this result or by another means to show that for all β,γ E R

    s(β+γ)=s(β)c(γ)+s(γ)c(β)


    Solution:

    It is difficult for me to write it out on here because I don't know how to write all the maths properly like everybody else on maths forum because I am new!

    but for A) I think I must be going wrong because for both s(x) and c(x) the ratio test is telling me they are both null series and thus they converge for all x. Is this correct?

    b) I am unsure how to begin this part, maybe i need to use part a)

    any advice would be good as I'm having trouble even really starting this question!

    Thank you.
    Just note that if f_n(x),g_n(x) denote the first and second sequence of functions then \left|f_n(x)\right|,\left|g_n(x)\right|\le \frac{n!}{(2n+1)!} for sufficiently large n. Thus the answer to the first is clearly all real x. And the second follows from Weirstrass.
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