Complete lift of vector fields and sprays to $T^\infty M$
In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in {\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of $M$. Then the concepts of vertical and complete lifts for functions and vector fields on $T^rM$ are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply $T^\infty M$ with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on $M$, can be lifted to a vector field or (semi)spray on $T^\infty M$. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on $T^\infty M$ including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite dimensional case of manifold of closed curves.