# Fourier analysis: Sine and Cosine series

I've been asked to explain the convergence of two series.

For $$x \in [0,\pi]$$:

$$\frac{sin(x)}{3}-\frac{sin(3x)}{1\cdot 3\cdot 5}-\frac{sin(5x)}{3\cdot 5\cdot 7}-\frac{sin(7x)}{5\cdot 7 \cdot 9}... = \frac{\pi}{8}sin^2(x)$$

And for $$x \in [-\frac{\pi}{2},\frac{\pi}{2}]$$:

$$\frac{cos(x)}{3}+\frac{cos(3x)}{1\cdot 3\cdot 5}-\frac{cos(5x)}{3\cdot 5\cdot 7}+\frac{cos(7x)}{5\cdot 7 \cdot 9}-... = \frac{\pi}{8}cos^2(x)$$

I took a crack (at the first one) by simply Fourier expanding $$\frac{\pi}{8}sin^2(x)$$ on the interval [0,$$\pi$$], but the coefficients come out wrong. I calculated them using the half-series Foruier expansion.

EDIT: I managed to get the first one right. The $$a_n$$'s are all zero, and the $$b_n$$'s are $$\frac{-1+(1)^n}{2n^3-8n}$$ which correctly yields $$1/3...-1/15...-1/105...$$.

For the second part I need the even expansion of $$cos^2(x)$$. However, simply calculating $$\frac{2}{\pi} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\pi}{8}cos^2(x) \cdot cos(2 n x)$$ yields all coefficients $$a_n$$ as zero. Which is obviously not right.

Furthermore: I am also asked to determine whether the two series converge pointwise, uniformly, or with $$L^2$$-convergence.

Any help much appreciated; thanks for reading.

Hint: if $$\sum_n |a_n| <\infty$$ then $$\sum _n a_n \sin (nx)$$ and $$\sum _n a_n \cos (nx)$$ are both absolutely and uniformly convergent by M-test. Verify that this property holds in your case.
• PS: uniform convergence also gives $L^{2}$ convergence of the Fourier series. – Kabo Murphy Oct 18 at 23:38
• Thanks for commenting, but at this point I'm not even sure how to prove that the series do converge to $\frac{\pi}{8}cos^2(x)$, as I said; the Fourier expansion came out wrong... For the test, I'll check if the infinite sum of $(|a_n| + |b_n|) < \infty$, but I have yet to show that I've gotten the right $a_n$ and $b_n$ (though I'm fairly confident about the $a_n$'s being zero) – Woodenplank Oct 19 at 13:37