Hello,
Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.
What is the problem with these? The third line in each tells you that the function is periodic with period
$\displaystyle 2 \pi$ so any interval of that length is what you use in the integrations and the basis functions are
$\displaystyle \sin(nx)$ and $\displaystyle \cos(nx)$.
RonL
Using integration by parts with $\displaystyle u=(\pi-t)$ and $\displaystyle dv=\cos(nt)$:
$\displaystyle \int_0^\pi (\pi-t) \cos(nt)~dt = \left[ \frac{(\pi-t)}{n} \sin(nt) \right]_0^{\pi}+\int_0^{\pi}\frac{1}{n^2} \sin(nt)~dt$$\displaystyle =\frac{1}{n^2}(1-(-1)^n)$
which is $\displaystyle 2/n^2,\ n=1, 3, 5, ..$ and $\displaystyle 0,\ n=2,4,.. $
RonL