How to show $\lim\limits_{n \to \infty} \int_0^{2\pi}\cos(nx)f(x)dx=0$ using integration by parts?
Prove if $f∈ C^1[0, 2\pi], \lim\limits_{n \to \infty} \int_0^{2\pi}\cos(nx)f(x)dx=0$
I understand there is a very similar question here, but I am specifically trying to use integration by parts.
Solution 1:
If you integrate by parts and differentiate $f$ you wind up with terms involving $f'(x)$ and $\dfrac{\sin(nx)}{n}$. The latter tends to zero uniformly.