Given the function $$f(x) = \ln\left(\frac{1+x}{1-x}\right)$$ Show that the error $f(1/3) - T_n(1/3)$ is at most $55/7776$
My attempt
Remainder Term = $[f^{(5)}(x) = 24/(1+x)^5 + 24/(1-x)^5]x^5/5!$
I am having difficulty in putting a limit on the $f^{(5)}(x)$ term so that I can calculate the maximum possible error....any tricks I can use here?
HINT: The remainder term is given by $f^{(5)}(c)\cdot(\frac{1}{3})^5/5!$ $\;\;$ where $0<c<\frac{1}{3}$ and
$f^{(5)}(c)=24\big(\frac{1}{(1+c)^5}+\frac{1}{(1-c)^5}\big)\le24\big(1+\frac{1}{(2/3)^5}\big)$.