Answer to Question #323297 in Real Analysis for alei

Question

What is the Fourier cosine series for f(x) = 2-π, in the interval 0 ≤ x < 2π?

Answer 1

$We\,\,have\\f\left( x \right) =2-\pi \\a_0=\frac{2}{2\pi}\int_0^{2\pi}{f\left( x \right) dx}=\frac{2\pi \left( 2-\pi \right)}{\pi}=2\left( 2-\pi \right) \\a_n=\frac{2}{2\pi}\int_0^{2\pi}{f\left( x \right) \cos \frac{nx}{2}dx}=\frac{1}{\pi}\left( 2-\pi \right) \int_0^{2\pi}{\cos \frac{nx}{2}dx}=\\=\frac{2-\pi}{\pi}\cdot \frac{2}{n}\sin \frac{nx}{2}|_{0}^{2\pi}=0\\f\left( x \right) =\frac{a_0}{2}=2-\pi$