Question

高等数学，1.求幂级数Σ（x^n）/（n+1） n=(0，∞)的和函数，2.尤其是当x=-1时的和函数请大神赐教。

Answer 1

简单计算一下即可，答案如图所示

Answer 2

S(x) = ∑<n=0,∞> x^n/(n+1) = (1/x)∑<n=0,∞> x^(n+1)/(n+1) (x≠0)
S(x) = 0 (x=0)
记 G(x) = ∑<n=0,∞> x^(n+1)/(n+1),
则 G'(x) = ∑<n=0,∞> x^n = 1/(1-x), (-1<x<1)
得 G(x) = ∫<0,x> dt/(1-t) = -ln(1-x), (-1≤x<1)
S(x) = -ln(1-x)/x (-1≤x<1)
S(-1) = ln2

追问

G'(x) = ∑ x^n = 1/(1-x), (-1 dt/(1-t) = -ln(1-x), (-1≤x<1)
微分在-1点处无意义，为何积分后在-1点处就有意义了呢，大神能解释解释么

追答