Question

求大神帮忙解答啊

Answer 1

设级数Un=n*x^n
Un+1=(n+1)x^(n+1)
比值法
lim n→∞ |Un+1/Un|
=lim n→∞ |(n+1)x^(n+1)/n*x^n|
=lim |x|(n+1)/n
=|x|＜1
收敛区间(-1，1)
设S(x)=∑n x^n
S(x)/x=∑n x^(n-1)
f(x)=∑n x^(n-1)
积分得
∫f(x)dx=∑x^n +C
等比级数求和，或者直接代公式
∑x^n=x/(1-x)
然后再求导
f(x)=[x/(1-x)]'=(1-x+x)/(1-x)²=1/(1-x)²
则S(x)=xf(x)=x/(1-x)²
将x=-1/3代入得
S(-1/3)=(-1/3)/(1+1/3)²=(-1/3)/(16/9)=-3/16