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Series representation help

  1. Jul 24, 2009 #1
    Does anyone know of a series representation for:

    [tex]\frac{sin(x)}{cos(x)+cosh(x)}[/tex]

    Preferably valid for 0<x, but any ideas or assistance on any domain would be much appreciated.
     
  2. jcsd
  3. Jul 24, 2009 #2
    Do a taylor series, for sin(x), cos(x), cosh(x), 1/x, then compose them and use the multinational theorm.
     
  4. Jul 24, 2009 #3

    zcd

    User Avatar

    [tex]\sin(x)=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}[/tex]
    [tex]\cos(x)=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...[/tex]
    [tex]\cosh(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+...[/tex]

    You could cancel some of the terms on denominator to get [tex]1+\frac{x^{4}}{4!}+\frac{x^{8}}{8!}+...=\sum_{n=0}^{\infty}\frac{x^{4n}}{(4n)!}[/tex] and then do long division.

    Wolfram Alpha gives
    [tex]\frac{x}{2}-\frac{x^{3}}{12}+\frac{x^{5}}{60}-\frac{17x^{7}}{5040}+\frac{31x^{9}}{45360}-\frac{691x^{11}}{4989600}+...[/tex]
     
    Last edited: Jul 24, 2009
  5. Jul 25, 2009 #4
    And Maple disagrees in some signs...
    [tex]{\frac {\sin \left( x \right) }{\cos \left( x \right) +\cosh \left( x
    \right) }} = {\frac {1}{2}}x-{\frac {1}{12}}{x}^{3}-{\frac {1}{60}}{x}^{5}+{\frac
    {17}{5040}}{x}^{7}+{\frac {31}{45360}}{x}^{9}-{\frac {691}{4989600}}{x
    }^{11}+O \left( {x}^{12} \right)
    [/tex]
     
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