Find the MacLaurin series for 11xSolutionthe derivatives of

Solution

the derivatives of this function form an interesting pattern...

starting with f=(1-x)^(-1), you have that f\'=-1x(1-x)^(-2)x)(-1)=
2/(1-x)^2

differentiating this gives you -2x(1-x)^(-3)(-1) = 3/(1-x)^3

the nth derivative of this function is n/(1-x)^(n+1)

thus, the value of f^n(0)=1

so the Maclaurin expansion becomes

f(x)=f(0)+f\'(0)x+2f\"(0)x^2/2! + 3f\'\'\'(0)x^3/3!+...

and f(x)=1+X+x^2+x^3+...