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