Fourth inverse-power sum of the cantilever spectrum (${\sum }_{n\ge 1}{q}_n^{-4}=1/12$)

For the positive roots $q_n$ of $\cos q\cosh q=-1$, ${\sum }_n{q}_n^{-4}=1/12$. The tip-force compliance of a cantilever is $L^3/(3A)$ directly, whereas expansion in normalized clamped--free bending modes and $W_n(L{)}^2=4L^{-1}\int W_n^{2}dx$ gives $4L^3{\sum }_n{q}_n^{-4}/A$. Equating the two responses proves the sum and evaluates the full thermal endpoint variance.