收敛半径 limn->∞Cn/Cn+1=Cn/(Cn+1/(n+1))=limn->∞1-1/(n+1)*1/(Cn+1/(n+1))
当 n -∞ Cn 大于等于1 1/(n+1)=0 即 1/(Cn+1/(n+1)) 为有界数
1/(n+1) = 0
则收敛半径为 R=1
当收敛时 即 |x|∞∑(n=1,∞)（1+1/2+1/3+···+1/n）x^n
=limn->∞ 1*x^1 +(1+1/2)x^2+.+(1+1/2.+1/n)x^n
=limn->∞ (x^1+x^2+...+x^n)+1/2(x^2+x^3+.+x^n)+.1/n*x^n
=limn->∞ (x-x^n)/(1-x)+1/2(x^2-x^n)/(1-x)+...+1/n*x^n
=limn->∞ (1*x+1/2*x^2+...+1/n*x^n)/(1-x)+(1+1/2+...+1/(n-1))*x^n/(1-x)
=limn->∞ (1*x+1/2*x^2+...+1/n*x^n)/(1-x)={limn->∞ (1*x+1/2*x^2+...+1/n*x^n)}/(1-x)
令f(x)=limn->∞ (1*x+1/2*x^2+...+1/n*x^n)
则 f'(x)=limn->∞(1+x+...+x^(n-1))=limn->∞(1-x^(n-1))/(1-x)=1/(1-x)
则 f(x)=-ln(1-x)+C 即
f（0）=C=limn->∞ (1*x+1/2*x^2+...+1/n*x^n)=limn->∞ (1+1/2+...+1/n)*0=0
limn->∞∑(n=1,∞)（1+1/2+1/3+···+1/n）x^n=-ln(1-x)/(1-x)
和函数为-ln(1-x)/(1-x)