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Math Help - Root of a Power Series

  1. #1
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    Root of a Power Series

    I have to find the "second smallest root" of the following equation :



    Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1]

    p =
    0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000
    >> roots(p)
    ans =
    35.5690
    -4.6796 +18.5352i
    -4.6796 -18.5352i
    4.1776 + 3.2154i
    4.1776 - 3.2154i
    1.4350

    But I have been asked to identify this series as well, which I am unable to do. Can anybody help me identify this series as a function or a product of functions? Thanking anybody who answers before hand
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  2. #2
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    Are you asking about roots for the infinite power series equation? Roots of the 6th degree polynomial formed by the first 7 terms do not tell you anything about roots of the entire series. This looks like the power series for e^{-x} which is never 0.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    [snip]
    This looks like the power series for e^{-x} which is never 0.
    The only problem with that observation being that the denominator of each coefficient has the form (n!)^2 rather than n! ....
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  4. #4
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    AS Mr. Fanatstic pointed out the power series of e^x is 1+x+(x^2)/(2!)+(x^3)/(3!)+...

    The problem with this series is the square of the nth factorial is involved.
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  5. #5
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    Quote Originally Posted by sparklingway View Post
    I have to find the "second smallest root" of the following equation :



    Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1]

    p =
    0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000
    >> roots(p)
    ans =
    35.5690
    -4.6796 +18.5352i
    -4.6796 -18.5352i
    4.1776 + 3.2154i
    4.1776 - 3.2154i
    1.4350

    But I have been asked to identify this series as well, which I am unable to do. Can anybody help me identify this series as a function or a product of functions? Thanking anybody who answers before hand
    I believe the function you seek is

    J_0(2\sqrt{x})

    where J_0 is a Bessel function of the first kind. If correct, the second smallist zero I got using Maple is

    x = 7.61781559
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  6. #6
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    Thanks. I got the problem solved yesterday
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