How do you find the power series for $f(x)=(2x)/(1+x^4)$ and determine its radius of convergence?

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Answer 1 · 2017-10-17T22:22:34

$(2x)/(1+x^4) = 2sum_(n=0)^oo(-1)^nx^(4n+1)$

for $absx < 1$

$sum_(n=0)^oo q^n = 1/(1-q)$

which converges for $absq < 1$ .

Substituting $q=-x^4$ we have:

$1/(1+x^4) = sum_(n=0)^oo (-x^4)^n = sum_(n=0)^oo (-1)^nx^(4n)$

converging for $absx^4 < 1$ , that is for $abs x < 1$ .

Multiplying by $2x$ term by term we finally get:

$(2x)/(1+x^4) = 2sum_(n=0)^oo(-1)^nx^(4n+1)$

How do you find the power series for f(x)=(2x)/(1+x^4)f(x)=(2x)/(1+x^4) and determine its radius of convergence?