This is my first question posted so sorry if formatting is not ideal. There are a few topics which I'm sketchy about before my applied math exam and would like additional sources for this topics. One of them is this idea of analysing the series solution for Sturm-Liouville and determining convergence. Consider the Fourier series expansion of $f(x)=x$ for $\pi<x<\pi$:$$x =2\sum_{n=1}^\infty\frac{(-1^{n+1})}{n}\sin(nx)$$ As $n\to \infty$ the series converges like $\frac{1}{n}$ and by the Riemann Lebesgue Lemma the periodic extension of this function is discontinuous and exhibits Gibbs phenomena at the boundary. Attempting to differentiate this Fourier Series expansion of $x$ would result in a series that would not converge. However, we can use this representation of $x$ to accelerate the convergence of solutions of Sturm-Liouville problems. Consider: $$S(x) = \sum_{n=2}^\infty(-1)^n\frac{n^3}{n^4+1}\sin(nx)$$ Which is the solution of the some P.D.E. The series goes down like $\frac{1}{n}$ for large n but we can use the previous Fourier series expansion of x given above and extract it out of this solution so we get a solution which converges more rapidly and has a continuous periodic extension. The details of this method are outlined in the attachment. If anyone can give me a better explanation of this procedure and any outside resources or problems that I can do, that would be much appreciated. Thanks!
