Discovering Coefficients of f(x)=5xarctan(3x) Power Series

In summary, the conversation discusses finding the first few coefficients in the power series representation of the function f(x)=5xarctan(3x). The homework equations state that the power series is represented in the form sum(Cn*x^n), and the attempt at a solution involves writing the function as an integral and splitting 5x with the trigonometric component. The individual asking the question is unsure of their approach and is seeking clarification. The response suggests using the Taylor series of tan^{-1}x at \alpha=0 and replacing x with 3x while multiplying by 5x after taking the derivative.
  • #1
rocksteady
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0

Homework Statement



The function [tex]f(x)=5xarctan(3x)[/tex] is represented as a power series. Find the first few coefficients in the power series.

Homework Equations



The power series is represented in the form [tex]sum(Cn*x^n)[/tex]

The Attempt at a Solution



I've attempted to write the function as an integral, and split the 5x with the trignonometric component. After doing this (and writing a power series), I'm not getting the right answer. Is something wrong with my approach?

Thank you in advance.
 
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  • #2
Take f(x)=tan[itex]^{-1}[/itex]x and find the taylor series of it at [itex]\alpha=0[/itex] and the replace x by 3x and multiply by 5x after
 

Related to Discovering Coefficients of f(x)=5xarctan(3x) Power Series

What is a power series?

A power series is a mathematical representation of a function as an infinite sum of terms, where each term is a polynomial multiplied by a constant.

How do you find the coefficients of a power series?

To find the coefficients of a power series, you can use the formula for the nth derivative of a function multiplied by x^n and evaluate it at x=0. In other words, you take the nth derivative of the function, multiply it by x^n, and substitute x=0 into the resulting expression.

What is the formula for the coefficients of f(x)=5xarctan(3x) power series?

The formula for the coefficients of f(x)=5xarctan(3x) power series is given by:
cn = (1/3)^n * (-1)^n * 5 * (2n)! / (n!(2n+1)!)
where cn represents the coefficient of x^n term in the power series.

Can the coefficients of a power series be negative?

Yes, the coefficients of a power series can be negative. In fact, in the formula for the coefficients of f(x)=5xarctan(3x) power series, the (-1)^n term ensures that the coefficients alternate between positive and negative values.

How do you use the coefficients to approximate the function?

To approximate the function using the coefficients, you can use the first few terms of the power series and add them together. The more terms you use, the more accurate the approximation will be. You can also use a calculator or computer program to calculate the value of the function at a specific point by plugging in the coefficients and the desired value of x into the power series formula.

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