Series Representation for sin(x)/(cos(x)+cosh(x)) Valid for 0<x?

In summary, to find a series representation for \frac{sin(x)}{cos(x)+cosh(x)}, you can use the Taylor series for sin(x), cos(x), and cosh(x) and then cancel out terms in the denominator. Wolfram Alpha and Maple may give slightly different results.
  • #1
rman144
35
0
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.
 
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  • #2
Do a taylor series, for sin(x), cos(x), cosh(x), 1/x, then compose them and use the multinational theorm.
 
  • #3
[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:
  • #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]
 

Related to Series Representation for sin(x)/(cos(x)+cosh(x)) Valid for 0<x?

1. What is a series representation?

A series representation is a way of expressing a mathematical function as an infinite sum of simpler functions. It is often used in calculus and other areas of mathematics to approximate functions that are difficult to evaluate directly.

2. How do I find the series representation of a function?

To find the series representation of a function, you can use techniques such as Taylor series or Fourier series. These methods involve using derivatives or trigonometric functions to express the function as an infinite sum.

3. Why is series representation important?

Series representation is important because it allows us to approximate complicated functions with simpler ones. This can make it easier to evaluate the function or perform calculations with it. Series representation also has many applications in physics, engineering, and other fields.

4. Can series representation be used for any function?

No, series representation is not always possible or practical for every function. It depends on the properties of the function and the techniques used to find the series representation. Some functions may not have a series representation, while others may require very complex series that are difficult to work with.

5. How accurate is series representation?

The accuracy of series representation depends on the number of terms used in the series. Generally, the more terms included, the more accurate the approximation will be. However, for some functions, an infinite number of terms may be needed for perfect accuracy. It is important to carefully consider the convergence of the series when using series representation for calculations.

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