母函数：1/(1-x)=1+X+X^2+.是怎么推导出来的?如题?

Question

母函数：1/(1-x)=1+X+X^2+.是怎么推导出来的?
如题?

Answer 1

f(x) =1/(1-x)
f'(x) = 1/(1-x)^2
f^(n)x = n!/(1-x)^n
f^(n)(0)/n!= 1
f(x) = f(0) +[f'(0)/1!]x+[f''(0)/2!]x^2+[f'''(0)/3!]x^3+...
= 1+x+x^2+.