将函数ln√[（1+x）/（1-x）]展成x的幂级数,并指明收敛区间.

题目

将函数ln√[（1+x）/（1-x）]展成x的幂级数,并指明收敛区间.

答案

f(x) = ln√[(1+x)/(1-x)] = (1/2) ln(1+x) ﹣ (1/2) ln(1-x)f '(x) = (1/2) [ 1/(1+x) ﹣1/(x-1) ] = 1/(1-x²) = ∑(n=0:∞) x^(2n),收敛区间（-1,1）积分,即得：f(x) = ∑(n=0:∞) x^(2n+1)/(2n+1) ,收敛区间...