Geodesic deviation to all orders via a tangent bundle formalism

We establish an in-in formalism for geodesic deviation as an alternative to Synge calculus, based on a covariant calculus of differential forms in tangent bundle. This derives the exact Lagrangian and equations governing the finite geodesic deviation between a free-falling test particle and an arbitrary observer, in terms of infinite sums whose coefficients are products of binomial coefficients. Explicit expressions are provided up to tenth order, finding agreements with the previous fourth-order result.