∑(x^2n)*(-1)^(n-1)/n(2n-1)

设f(x)=∑(1->∞)(x^2n)*(-1)^(n-1)/n(2n-1)=∑(1->∞)(x^2n)*(-1)^(n-1)/n(2n-1)
f'(x)=∑(1->∞)(2x^(2n-1))*(-1)^(n-1)/(2n-1)
f''(x)=∑(1->∞)(2x^(2n-2))*(-1)^(n-1)
=2∑(0->∞)(x^(2n))*(-1)^n
=2∑(0->∞)((x^2)^n))*(-1)^n=2∑(1->∞)(-x^2)^n)
=2(1+-x^2+x^4-x^+...)
=2/(1+x^2)
f'(x)=∫(-x^2)/(1+x^2)= 2arctanx
f(x)= -log(x^2+1)+2 x tan^(-1)(x)