How can I create a power series expansion for x^2 / (1+x^2)?

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SUMMARY

The discussion focuses on creating a power series expansion for the function x² / (1+x²). Participants suggest utilizing the geometric series approach, which is effective for this type of function. Additionally, leveraging the known series for arctan(x) and its derivative, 1/(1+x²), is recommended as an alternative method. Participants emphasize the importance of manipulating the fraction to fit the standard geometric series form, 1/(1-r), to facilitate the expansion process.

PREREQUISITES
  • Understanding of power series expansions
  • Familiarity with geometric series
  • Knowledge of the arctan(x) series
  • Basic calculus concepts, particularly derivatives
NEXT STEPS
  • Study the derivation of the geometric series formula
  • Learn how to manipulate fractions to fit the geometric series form
  • Explore the series expansion of arctan(x) and its applications
  • Practice deriving power series for various rational functions
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Mathematicians, students studying calculus, and anyone interested in series expansions and their applications in mathematical analysis.

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I was wondering how to create a power series expansion for the function
x^2 / (1+x^2)... I've tried using the geometric series, but somehow i got stuck.
thanks.
 
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Geometric series should work fine. How about you post your work?

An alternate method: if you know the series for arctan(x), you can use the fact that it's derivative is 1/(1+x^2).
 
It may also help you to think in what form geometric series are usually in, i.e.

1/(1-r). So maybe altering your fraction a bit will help you notice it.
 

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