求级数1/2^n(2n+1)的和

Question

求级数1/2^n(2n+1)的和

Answer 1

令 S(x)=∑ x^(2n)]/(2n+1)
两边同乘x
xS(x)=∑ x^(2n+1)]/(2n+1)是时求导
（xS(x)）的导数=∑x^2n=1/(1-x^2) 然后两个各分
xS(x)＝1/(1-x^2)的积分＝(1/2)ln[(1+x)/(1-x)]
s(x)=(1/2x)ln[(1+x)/(1-x)] x不为0
1 x＝0
令x=(根号2 ) /2代入s(x)即为所求