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x = 0 时 ， S(x) = ∑<n=1,∞> [1/(2n+1) - 1] x^(2n) = 0.
x ≠ 0 时， S(x) = ∑<n=1,∞> [1/(2n+1) - 1] x^(2n)
= ∑<n=1,∞> x^(2n)/(2n+1) - ∑<n=1,∞> x^(2n)
= (1/x)∑<n=1,∞> x^(2n+1)/(2n+1) - x^2/(1-x^2)
= (1/x)S1(x) - x^2/(1-x^2)
[S1(x)]' = ∑<n=1,∞> x^(2n) = x^2/(1-x^2)
S1(x) = ∫<0+, x> [t^2/(1-t^2)]dt
= (1/2)ln[(1+x)/(1-x)] - x
S(x) = [1/(2x)]ln[(1+x)/(1-x)] - 1/(1-x^2) (x ≠ 0)
S(x) = 0 (x = 0)
x ≠ 0 时， S(x) = ∑<n=1,∞> [1/(2n+1) - 1] x^(2n)
= ∑<n=1,∞> x^(2n)/(2n+1) - ∑<n=1,∞> x^(2n)
= (1/x)∑<n=1,∞> x^(2n+1)/(2n+1) - x^2/(1-x^2)
= (1/x)S1(x) - x^2/(1-x^2)
[S1(x)]' = ∑<n=1,∞> x^(2n) = x^2/(1-x^2)
S1(x) = ∫<0+, x> [t^2/(1-t^2)]dt
= (1/2)ln[(1+x)/(1-x)] - x
S(x) = [1/(2x)]ln[(1+x)/(1-x)] - 1/(1-x^2) (x ≠ 0)
S(x) = 0 (x = 0)
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