求幂级数∑（上面∞,下面n＝0）x的n次方/n+1的和函数,

Question

求幂级数∑（上面∞,下面n＝0）x的n次方/n+1的和函数,

Answer 1

∑（0,∞）x^n/(n+1)
=1/x*∑（0,∞）x^(n+1)/(n+1)
令f(x)=∑（0,∞）x^(n+1)/(n+1)
则f'(x)=∑（0,∞）x^n=1/(1-x)
两边对x积分得
f(x)=-ln(1-x)
故和函数为-ln(1-x)/x,|x|