The Fourier series of the function $f ( x ) = x$ on $[ - 1,1 ]$ is
A) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 2 \cos ( n \pi x ) / ( n \pi )$
B) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \cos ( n \pi x ) / ( n \pi )$
C) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 \sin ( n \pi x ) / ( n \pi )$
D) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \sin ( n \pi x ) / ( n \pi )$
E) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \sin ( n \pi x ) / ( n \pi )$

Question

Quizplus · Accepted Answer

The Fourier series of an odd function $f(x) = x$ on the interval $[-1, 1]$ consists of sine terms only, because sine functions are odd and the function $f(x) = x$ is also odd. The correct series uses sine terms and alternates in sign, which matches option D.

The Fourier Series of the Function f(x)=xf ( x ) = xf(x)=x

The Fourier Series of the Function $f ( x ) = x$