Problem: Approximate
\begin{equation} \int_0^{0.5}\dfrac{x}{1+x^5}\,dx \end{equation}
using three terms of the series expansion and approximate the remainder.
Work:
\begin{eqnarray} \dfrac{x}{1+x^5}&\approx\sum_{n=0}^2(-1)^nx^{5n-1}\\ \int_0^{0.5}\sum_{n=0}^2(-1)^nx^{5n-1}\,dx&=\left[\displaystyle\sum_{n=0}^2\dfrac{(-1)^nx^{5n+2}}{5n+2}\right]_0^{0.5}\\ &=\displaystyle\sum_{n=0}^2\dfrac{(-1)^n\left(\frac{1}{2}\right)^{5n+2}}{5n+2}\\ &=\dfrac{\left(\frac{1}{2}\right)^2}{2}-\dfrac{\left(\frac{1}{2}\right)^7}{7}+\dfrac{\left(\frac{1}{2}\right)^{12}}{12}\\ &=\frac{1}{8}-\frac{1}{896}+\frac{1}{49152}\\ &=0.124 \end{eqnarray}
How do I estimate the remainder?
I am pretty confident I did the integral correctly and calculated the first 3 terms but I am not sure how to find the remainder? Do I use the integral test?