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.

  • $\begingroup$ PS: uniform convergence also gives $L^{2}$ convergence of the Fourier series. $\endgroup$ – Kabo Murphy Oct 18 at 23:38
  • $\begingroup$ 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) $\endgroup$ – Woodenplank Oct 19 at 13:37

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