The error for the Gaussian quadrature formulae of order $N$ is given by $$\begin{equation*} \int_a^bW(x)f(x)dx-\sum_{k=1}^Nw_kf(x_k)=\frac{f^{2N}(\xi)}{(2N)!}\int_a^bW(x)[q_{N}(x)]^2dx \end{equation*}$$ where $q_{N}$ is the chosen orthogonal polynomial and $\xi$ is a number in the interval $[a,b]$. We have assumed that $f\in C^{2N}[a,b]$, viz. the space of all real or complex $2N$ times continuously differentiable functions.