Question #40709

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that no two are orthogonal. A sequence of vectors is defined as $\vec{v}_k = x_k (\vec{a} \times (\vec{b} \times \vec{c})) + y_k (\vec{b} \times (\vec{c} \times \vec{a})) + z_k (\vec{c} \times (\vec{a} \times \vec{b}))$ where $x_k = \frac{4(k^2+k-1)}{(k+2)!}$, $y_k = \frac{2^k(k-1)}{(k+1)!}$, and $z_k = \frac{\lambda}{k(k+1)(k+2)}$. If $\sum_{k=1}^\infty \vec{v}_k = \vec{0}$, then the value of $\lambda$ is