The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and eigenfunctions explicitly.

$$\frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0$$

On DLMF there are eigenvalue expansions for $|q|<<1.$ Unfortunately, no error estimates are given. Thus, I was wondering if anybody here knows a way to bound the error for given $q$ if I only take the first two terms.

So, I want to know how to bound the error between $|a_i(q)-i^2 + \text{second term}(q,q^2)|.$

Are there rigorous methods of perturbation theory that apply here?