Question

已知幂级数,求和函数

Answer 1

2S(x) = ∑2nx^(2n)

= ∑(2n+1)x^(2n) - ∑x^(2n)

=[∑x^(2n+1)]' - ∑x^(2n)

=[x^3/(1-x^2)]' - x^2/(1-x^2)

=(3x^2-x^4)/(1-x^2)^2-(x^2-x^4)/(1-x^2)^2

=2x^2/(1-x^2)^2.

则 S(x)=x^2/(1-x^2)^2.

收敛域 -1