Ramp-state expansion in a symmetric infinite well ($c_n=-2\sqrt{3}(-1{)}^n/(n\pi )-4\sqrt{3}\sin (n\pi /2)/(n^2{\pi }^2)$)

In a well $-a<x<a$, the normalized state proportional to $x$ on $0<x<a$ and zero elsewhere has stationary-state coefficientsThey follow from the orthonormal basis inner products. Their squared magnitudes give energy-measurement probabilities by the Born rule; splitting even and odd indices in the Parseval identity gives convergent integer-coefficient series identities. The expansion is interpreted in the square-integrable state norm.