将y=(1+x)ln(1+x)展成麦克劳林级数,

Question

将y=(1+x)ln(1+x)展成麦克劳林级数,

Answer 1

f(x)=(1+x)ln(1+x) f(0)=0
f'(x)=ln(1+x)+1 f'(0)=1
f''(x)=(1+x)^-1 f''(0)=1
f'''(x)=(1+x)^-2 f'''(0)=1
.
f^(n)(x)=(1+x)^-(n-1) f^(n)(0)=1
y=(1+x)ln(1+x)=
f(0)+f'(0)x+f''(0)/2!x^2+...+f^(n)(0)x^n=
x+x^2/2!+x^/3!3+x^4/4!+...+x^n/n!+...