How to Show that Coefficients of a Power Series for tan(x) are 0 for Even n

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SUMMARY

The coefficients of the power series for tan(x) are zero for all even n due to the odd function property of tan(x). Specifically, since tan(-x) = -tan(x), any non-zero coefficients for even-powered terms would contradict this property. Therefore, it is established that An = 0 for even n in the power series expansion centered at x=0.

PREREQUISITES
  • Understanding of power series expansions
  • Knowledge of odd and even functions
  • Familiarity with the function tan(x)
  • Basic calculus concepts related to series and limits
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  • Study the properties of odd and even functions in detail
  • Explore power series expansions for other trigonometric functions
  • Learn about Taylor series and their applications
  • Investigate the convergence of power series at different points
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Mathematicians, students studying calculus, and anyone interested in the properties of trigonometric functions and their series expansions.

complexhuman
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Hi

Just had a question.

Assuming tan(x) is given by a power series with coefficiants (An). How can it be shown that An = 0 whenever n is even.



Thanks
 
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I assume you mean a power series centered about x=0. tan(x) is an odd function, i.e. tan(-x)=-tan(x). If it had nonzero coefficients to its even powered terms this could not be the case.
 
thanks a lot :)
 

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