Let \(\displaystyle $f$\) be a continuously differentiable function on the interval \(\displaystyle $[0,2\pi]$\), where \(\displaystyle $f(0) = f(2\pi)$\) and \(\displaystyle $f'(0) = f'(2\pi)$\). For \(\displaystyle $n = 1,2,3,\dotsc$\), define
\(\displaystyle \[
a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) \dd x.
\]\)
Prove that the series\(\displaystyle $\sum_{n=1}^\infty |a_n|^2$\) converges.
So far I've got that
\(\displaystyle \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) \dd x = \frac{1}{2\pi n}\int_0^{2\pi} f'(x)\cos(nx) \dd x\)
via integration by parts and the conditions on \(\displaystyle f\). I've also got that
\(\displaystyle |a_n| \le \frac{1}{2\pi} \int_0^{2\pi} |f(x)| \dd x\)
and
\(\displaystyle |a_n| \le \frac{1}{2\pi n} \int_0^{2\pi} |f'(x)| \dd x\)
Both from the boundedness of sine and cosine. I also know that \(\displaystyle f\) is a rectifiable curve, although I'm not sure this helps at all. I'm just stuck as to where to go from here, and whether the bounds on \(\displaystyle |a_n|\) are actually helping at all. If you could point me in the right direction I'd be very grateful.
Thanks.
Edit: I realized I left out an \(\displaystyle n\) in the denominator of the constant on the second bound of \(\displaystyle |a_n|\). This would imply that in the limit \(\displaystyle |a_n|^2\) tends to 0 so the series has at least a chance to converge.
\(\displaystyle \[
a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) \dd x.
\]\)
Prove that the series\(\displaystyle $\sum_{n=1}^\infty |a_n|^2$\) converges.
So far I've got that
\(\displaystyle \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) \dd x = \frac{1}{2\pi n}\int_0^{2\pi} f'(x)\cos(nx) \dd x\)
via integration by parts and the conditions on \(\displaystyle f\). I've also got that
\(\displaystyle |a_n| \le \frac{1}{2\pi} \int_0^{2\pi} |f(x)| \dd x\)
and
\(\displaystyle |a_n| \le \frac{1}{2\pi n} \int_0^{2\pi} |f'(x)| \dd x\)
Both from the boundedness of sine and cosine. I also know that \(\displaystyle f\) is a rectifiable curve, although I'm not sure this helps at all. I'm just stuck as to where to go from here, and whether the bounds on \(\displaystyle |a_n|\) are actually helping at all. If you could point me in the right direction I'd be very grateful.
Thanks.
Edit: I realized I left out an \(\displaystyle n\) in the denominator of the constant on the second bound of \(\displaystyle |a_n|\). This would imply that in the limit \(\displaystyle |a_n|^2\) tends to 0 so the series has at least a chance to converge.
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