Question

Find the nth maclaurin polynomial for the function. f(x) = 5x − 5 x + 1 , n = 4 p4(x)

Answer 1

Assuming the function is $f(x)=\dfrac{5x-5}{x+1}$, and not $5x-5x+1=1$ which would be its own Maclaurin polynomial right away.

$\dfrac{5x-5}{x+1}=\dfrac{5(x-1)}{x+1}=\dfrac{5(x+1-2)}{x+1}=5-\dfrac{10}{1+x}$

Recall the geometric power series,

$\displaystyle\sum_{n=0}^\infty x^n=\frac1{1-x}$

where $|x|$. Replace $x\mapsto-x$ and we have

$\displaystyle\sum_{n=0}^\infty (-x)^n=\frac1{1+x}$

and so

$f(x)=5-10\displaystyle\sum_{n=0}^\infty(-x)^n$

We want the 4th degree polynomial, so

$p_4(x)=-5+10x-10x^2+10x^3-10x^4$