For which $-2\pi < x < 2\pi$ does this series converge?
$$1 = \sum_n^{\infty} A_n\cos\left[\left(\frac{1}{2}+n\right)x\right]$$
The cosine function is piecewise orthogonal.
I found
$$A_n = \frac{(1, X_n)}{(X_n,X_n)} = \frac{4(-1)^n}{\pi(2n+1)}$$
which goes to $0$ as $n\to\infty$ which seems to imply that the series converges for all $x$ in the interval since $\max |X_n|= 1$ (the cosine function) .
Of course it won't always converge to $1$.
Is this correct or am I missing something?