Frobenius & Special Functions · Hard problem

Expand $f(x) = |x|$ on $[-1, 1]$ in Legendre polynomials, $|x| = \sum_{n\ge0}c_nP_n(x)$, where $P_2(x) = \tfrac12\left(3x^2 - 1\right)$. Find $c_2$. Express your answer as a common fraction.