Can the 'Almost Reimann Integral' be Recovered Using Pinsky's Textbook?

[confinement]

I am trying to show that a certain sum rigorously tends to a Reimann integral, but I have a certain sticking point that you will find below.

$$S = \sum_{n = 1}^{\infty} c_n \phi_n$$

$$\phi_n = \sqrt{\frac{2}{L}} \sin(\frac{n \pi x}{L})$$

In this case the c_n can be assumed to be such that the series converges. Basically I want to show that the limit of S as L goes to infinity is a Reimann integral by definition. We have:

$$\lim_{L\to\infty}\sum_{n = 1}^{\infty} \sqrt{\frac{2}{L}} c_n \sin(\frac{n \pi x}{L})$$

consider the function sin(k), and define a partition of the positive half of the real line by the points n pi / L. Then the mesh of the partition is pi / L. To deal with the constants define c_n = c(n pi x / L) = c(k_n). Then we have:

$$\lim_{L\to\infty}\sum_{n = 1}^{\infty} c(k_n)\sin(k_n) \sqrt{\frac{2}{L}}$$

This is almost a Reimann integral by definition, we have an infinite sum of values of a function multiplied by something small, and the mesh of the partition of points is going to zero, the problem is that the mesh of the partition is $\Delta k_n = \pi / L$, not the $\sqrt{\frac{2}{L}}$ that I have. The constant factor is obviously not a problem, but I cannot pretend that L and sqrt(L) are the same for this purpose. Is there anyway to recover, or does this 'almost integral' correspond to anything?

 

[olliemath]

Would it not be much easier to treat it like a Fourier series? :-) (Mind you, I haven't tried so I don't know)
Anywho I got a little confused when reading because you went from talking about sums of $$\sin(n\pi x/L)$$ terms to $$\sin(n\pi/L)$$, i.e. the value of the whole thing at 1. Anyway, assuming $$k_n=n\pi x/L$$ define $$c(k_n)=c_n\sqrt{n}$$, then each term in the sequence is

$$\frac{c(k_n) k_n^{-1/2}\sin(k_n)\sqrt{\pi x}}{L}$$

which should give you an integral.. proving its existence might be tricky.. but you pays your money you takes your choice.

 

[gammamcc]

Try the textbook by Pinsky: Partial Differential Equations and Boundary-Value Problems