Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 12 / c / Solution

The function $x^2-1$ is an even function, while every sine basis function is an odd function. Their product is odd, so every sine coefficient is zero:The function itself is not zero, and therefore it cannot be represented by this sine basis. The basis spans the odd subspace, whereas the requested nonzero even function lies in its orthogonal complement. Equivalently, any pointwise limit of odd partial sums remains odd wherever both reflected points have limits.