I have arrived at the following expression:
$$\sum_{n=0}^{\infty} \sqrt{n+1}D_{n}\left(x\right)$$
Where $D_n\left(x\right) = \frac{\sin\left(\left(n+\frac{1}{2}\right)x\right)}{\sin\left(x/2\right)}=1+2\sum_{k=1}^{n} \cos\left(kx\right)$ is the Derichlet kernel. I expected this expression to essentially be a Dirac delta function, but the connection seems a bit unclear to me. Does a sum of Derichlet kernels have any significance or direct relation to Dirac delta functions?
The only progress I've made is by using the second definition of the Derichlet kernel and re-arranging the terms in the double sum:
$$\sum_{n=0}^{\infty} \sqrt{n+1}D_{n}\left(x\right)=\sum_{n=0}^{\infty}\sqrt{n+1}\left[1+2\sum_{k=1}^{n} \cos\left(kx\right)\right]$$
Then if we truncate the series at $N$:
$$=\left(\sqrt{1}+\sqrt{3}+\dots\sqrt{2N+1}\right)1+\dots$$ $$\dots\left(\sqrt{1}+\sqrt{3}+\dots\sqrt{2\left(N-1\right)+1}\right)\cos\left(x\right)+\dots$$ $$\dots\left(\sqrt{1}+\sqrt{3}+\dots\sqrt{2\left(N-2\right)+1}\right)\cos\left(2x\right)$$
Which looks like:
$$\sum_{n=0}^{N} \sqrt{n+1}D_{n}\left(x\right)=\sum_{k=0}^{N} a_k\cos\left(kx\right)$$
with $a_k=2\sum_{m=0}^{N-k}\sqrt{2m+1}$
Which, being a cosine series seems sort of like it would describe a delta function, but its still a bit hand wavy.
