Q:

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need some refreshing, even for this textbook problem.  
Here is my approach, but I am almost certain something is amiss: Taylor's Theorem states that the $n$th remainder polynomial for the nth Taylor polynomial is $$R_n(x)= \frac{f^{(n+1)}(c) \ (x-a)^{n+1} }{(n+1)!} , $$ where $a$ is the center and the existence of $c \in [a,x]$ is guaranteed by the Mean Value Theorem.  I found the $4^{th}$ derivative of $\ln (1+x)$ to be $-\frac{6}{(x+1)^4}.$ The absolute error is given by $M\frac{|x-a|^{n+1}}{(n+1)!}$, where $|f^{(n+1)}(x)|≤M, x\in[0, .09]$. Since the $4^{th}$ derivative is just a hyperbola shifted to the left $1$ unit and "scaled" by a factor of $-6$, the graph has two branches with no inflection points, therefore only the endpoints of the interval $[0,.09]$ need be checked.  The value of $0$ gives the greatest magnitude, and so $$ |R_3(x)|= 6\frac{x^{3+1}}{(3+1)!}=\frac{x^4}{4} .$$ 
Evaluating this at $x=.09$ (because the original function was $\ln(x+1)$ and I am trying to estimate $\ln (1.09)$), we have $\frac{.09^4}{4} \approx 1.64 \times 10^{-5}$.
However, I am almost positive this is wrong.  What did I do wrong?  Any help would be greatly appreciated.  Thank you. 

A:

I do not know why you say your reasoning is wrong.  Anyway, the successive terms of the series expansions of $\log(1+x)$ have alternating signs when $x$ is positive.  If, furthermore, their amplitudes are decreasing, which is the case for the whole sequence of terms if $x$ is in $(0,1)$, Leibniz criterion yields rigorous upper and lower bounds.  For example, for every $x$ in $(0,1)$,
$$
x-\frac12x^2+\frac13x^3-\frac14x^4\leqslant\log(1+x)\leqslant x-\frac12x^2+\frac13x^3,
$$
hence the bound of your error term by $\frac14x^4$ is correct (and you get furthermore that it is one-sided).