Taylor polynomials of degree n

Question

I have this math question that states:

Find the Taylor polynomials of degree $n$ approximating $ln(1+x)$ for $x$ near $0$.

The $n$'s are 5, 7, and 9.

$f^{(5)}(0)=24$; I got the derivative to be $\frac{24}{(x+1)^5}$

$f^{(7)}(0)=720$; I got the derivative to be $\frac{720}{(x+1)^7}$

$f^{(9)}(0)=40320$; I got the derivative to be $\frac{40320}{(x+1)^9}$

Answer 1

A Taylor polynomial is a polynomial. Anything with $(x+1)^n$ in the denominator is not. The Taylor polynomial you are looking for is $f(0)+xf'(0)+\frac 1{2!}x^2f''(0)+\dots \frac 1{n!}x^nf^{(n)}(0)$ The three parts ask you to go out to $x^5, x^7,$ and $x^9$ respectively.