The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is given by $$\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n+3} = \frac{x}{3}-\frac{x^3}{5}+\frac{x^5}{7}-\cdots$$
I am trying to show that the error of the approximation of $g(1/2)$ by using only the first two (nonzero) terms in the series is less than $\frac{1}{200}$.
The approximation of $g(1/2)$ by using two terms is $\frac{\frac{1}{2}}{3}-\frac{\big(\frac{1}{2}\big)^3}{5}=\frac{17}{200}.$
Using the formula for the remainder when using $n$ terms: $$R_n=\frac{g^{(n+1)}(c)x^n}{(n+1)!}$$, where $c$ should be bwtween $0,1/2$, our error is $$|R_2|\le \frac{|g^{(3)}(c)|(\big(\frac{1}{2}\big)^2}{3!},$$
however, I am unable to find an upper bound of $g^{(3)}$ by directly computing the derivatives since it does not seem that the terms of $g^{(3)}$ are decreasing in magnitude (if they were decreasing in magnitude, I would use the first term as an upper bound since it is alternating).
How can I show that the error is at most $\frac{1}{200}$? I prefer a solution that does not find a closed formula for $g$.
For an alternating series that converges, the error using $n$ terms is less than the magnitude of the $(n+1)$th term.
This means you don't need to calculate any derivatives, and you can just use $$|R_2| < {\frac{|x^5|}{7}} < \frac{1}{32} . \frac{1}{7} < \frac{1}{200}$$ for $x=1/2$