find a power series for the following function f(x)=x/(1+x^2)

Expert Answers
sciencesolve eNotes educator| Certified Educator

You should remember that the geometric series `sum_(n=0)^oo x^n = 1/(1 - x)` , hence, you need to rewrite the function provided such that:

`f(x) = x*1/(1 - (-x^2))`

Hence, reasoning by analogy, yields:

`f(x) = x*sum_(n=0)^oo (-1)^n x^(2n)`

You should notice that the term x left outside will not influence the convergence of series.

Hence, evaluating a power series for the given function yields `x*sum_(n=0)^oo (-1)^n x^(2n).`