The Fourier series of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ is
A) $\sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right) + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
B) $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
C) $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
D) $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
E) $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$

Question

Quizplus · Accepted Answer

The Fourier series of $f(x) = x^2$ on $[-1,1]$ is an even function, so it only contains cosine terms and a constant term. The correct series includes the constant term $1/3$ and cosine terms, which matches option E.

The Fourier Series of the Function f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2

The Fourier Series of the Function $f ( x ) = x ^ { 2 }$