Fourth inverse-power sum of the cantilever spectrum
= Fourth inverse-power sum of the cantilever spectrum
{title2=$\sum_{n\geq1}q_n^{-4}=1/12$}
For the positive roots $q_n$ of $\cos q\cosh q=-1$, $\sum_nq_n^{-4}=1/12$. The <tip-force compliance of a cantilever> is $L^3/(3A)$ directly, whereas expansion in normalized <clamped--free bending modes> and $W_n(L)^2=4L^{-1}\int W_n^2dx$ gives $4L^3\sum_nq_n^{-4}/A$. Equating the two responses proves the sum and evaluates the full thermal endpoint <variance>.