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)#