Boundary Values & Fourier Series · Superchallenge problem

The eigenvalues $0 < \lambda_1 < \lambda_2 < \cdots$ of $y'' + \lambda y = 0$, $y(0) = 0$, $y'(1) + y(1) = 0$ are not expressible in closed form (they satisfy $\tan\sqrt\lambda = -\sqrt\lambda$). Nevertheless find $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{\lambda_n}$ exactly.