# Convergence of Unique Fourier Series

Hi, I'm new to this forum, so I apologize if my LaTeX looks messed up.

1. Find the Fourier Series for $$f(x) = \sqrt{|x|}$$ and prove it converges to $$f(x)$$

3. So, I've thus far proved that $$\sqrt{|x|}$$ is piecewise continuous by proving that the limit as x approaches 0 (from both the right and left) for $$f(x) = \sqrt{|x|}$$ is equal to 0. However, the function fails to be piecewise smooth because its derivative has a vertical tangent and, thus, most of the convergence theorems that I've learned (such as the Fourier series converging to the average of the right and left limits) are inaccessible.

So, I wrote out the Fourier series myself for the arbitrary interval -a < x < a, getting:

For Fourier coefficients, we get:

$$a_0 = \frac{1}{2a}\int_{-a}^a \sqrt{|x|} dx = \frac{2}{3}\sqrt{|x|}$$

$$a_n = \frac{2}{a}\int_0^a \sqrt{|x|} \cos{(\frac{n\pi x}{a})}$$

$$b_n = 0$$(because the function is even)

The Fourier series becomes

$$\sqrt{|x|} \approx \frac{2}{3}\sqrt{|x|} + \frac{2}{a}\sum_{n=1}^\infty \int_0^a \sqrt{|z|}\cos{(\frac{n\pi z}{a})} dz \cos{(\frac{n\pi x}{a})}$$

I tried to use a theorem stating that if the series $$\sum_{n=1}^\infty |a_n| + |b_n|$$ converges, then the Fourier series converges uniformly. So, essentially, I must prove
$$\sum_{n=1}^\infty |\frac{2}{a}\int_0^a \sqrt{|x|} \cos{(\frac{n\pi x}{a})}|$$ converges.

I tried to use an array of traditional convergence tests, but none of them seemed to work.

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