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

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SUMMARY

The function f(x)=5xarctan(3x) can be expressed as a power series by first finding the Taylor series of arctan(x) at α=0. The correct approach involves substituting x with 3x in the Taylor series of arctan(x) and then multiplying the resulting series by 5x. This method yields the coefficients for the power series representation of f(x). The initial coefficients can be derived from the first few terms of the resulting series.

PREREQUISITES
  • Understanding of Taylor series expansion
  • Familiarity with the arctangent function and its properties
  • Basic knowledge of power series representation
  • Ability to perform algebraic manipulations with series
NEXT STEPS
  • Study the Taylor series of arctan(x) in detail
  • Learn about power series convergence and radius of convergence
  • Explore techniques for manipulating power series
  • Practice deriving coefficients from power series using substitution methods
USEFUL FOR

Students studying calculus, mathematicians interested in series expansions, and educators teaching power series concepts.

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Homework Statement



The function f(x)=5xarctan(3x) 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 sum(Cn*x^n)

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.
 
Physics news on Phys.org
Take f(x)=tan^{-1}x and find the taylor series of it at \alpha=0 and the replace x by 3x and multiply by 5x after
 

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