This is a bit misleading. If the integrand is analytic, then the Gauss quadrature converges geometrically.

For example with f(x) = sqrt(1 + sin(x)) and you approximate integral f(x) dx from -1 to 1 with n Gauss points, then the the convergence is geometric:

    n | approx
    1 | 2.0------------- 
    2 | 1.9172----------
    3 | 1.917703--------
    4 | 1.917702153-----
    5 | 1.917702154417--
    6 | 1.91770215441681
   

But with a function that is not differentiable such as g(x) = |x|, there is definitely not geometric convergence.

      n | approx
      1 | 0.0-----------    
     10 | 1.007---------
    100 | 1.00008-------
   1000 | 1.0000008-----