Show that we can define a connection on any manifold using partitions of unity Suppose that $(U,\varphi)$ is a chart on manifold $M$, and $X,V$ are vector fields on manifold $M$, then we can write:
$$X=\sum_{i=1}^{i=n}X^{i}\frac{\partial}{\partial x^{i}}$$ on $U$, and define a connection on $U$ by: $$D_{V}X=\sum_{i=1}^{i=n}(VX^{i})\frac{\partial}{\partial x^{i}}\cdots\tag{1}$$
Let $\{U_{j}\}_{j=1}^{\infty}$ be a locally finite covering of $M$, where each $U_{j}$ is coordinate neighborhood on $M$. Let $D^{j}$ be the connection on $U_{j}$ defined by (1) respectively .
and let $\{f_{j}\}$ be the partitions of unity on $M$ that are subordinate to $\{U_{j}\}$.
Show that: $\sum_{j=1}^{\infty}f_{j}D^{j}$ is a connection on $M$.
Another question: suppose  $X,V$ are two vcetor fields on manifold $M$, with the connection $\sum_{j=1}^{\infty}$ $f_{j}D^{j}$ defined above, how do we know that the new vector field $(\sum_{j=1}^{\infty}f_{j}D^{j})_{V}X$ is well defined on some intersections of the coordinate neighborhood, say $U_{j_{1}}$ and $U_{j_{2}}$ .
 A: Let us write $D = \sum_{j=1}^{\infty}f_jD^j$. In order to check that $D$ is in fact a connection we need to verify three things:
1) Is it linear in $V$? That is, is it true that:
\begin{equation}
D_{g_1V_1+g_2V_2}X = g_1D_{V_1}X + g_2D_{V_2}X
\end{equation}
2) Does it satisfy the Liebniz law in $X$:
\begin{equation}
D_{V}fX = (Vf)X + fD_VX
\end{equation}
3) Is it linear over $\mathbb{R}$ in $X$?:
\begin{equation}
D_V(\alpha_1X_1+\alpha_2X_2) =\alpha_1D_VX_1 + \alpha_2D_VX_2
\end{equation}
where $\alpha_1,\alpha_2\in \mathbb{R}$
It is easy to show that each `local' connection $D^j$ on $U_j$ satisfies all three of these.
Now to show that $D$ satisfies $1$, $2$ and $3$, it will suffice to show that that it satisfies all three of these properties at each point $p$. So, choose a neighbourhood $W$ of $p$ such that $U_j\cap W = \emptyset$ for all but a finite number of $j$ (As per Jason De Vito's comment, we may do this if $\{U_j\}$ is a locally finite cover). Now if we want to calculate the value of $D_VX$ at $p$ it suffices to consider only the restrictions of $V$ and $X$ to $W$; $V|_{W}$ and $X|_{W}$. the hardest property to check is 2), so lets look at that:
\begin{align}
D_{V|_{W}}(gX_{W}) & = \sum_{j}f_jD^j_{V|_{W}}gX|_{W}\\
                   & = \sum_{j}(f_j(V_{W}g)X + gD^{j}_{V|_{W}}X) \\ 
                   & = (V_{W}g)X + gD_{V|_W}X \\
\end{align}
where in the second line we used the fact that each $D^j$ satisfies the Liebniz law while in the third line we are using the fact that for any partition of unity, $\sum_{j}f_j = 1$, and because  the sum restricted to $V$ is finite we avoid prickly issue of convergence. So our connection $D$ does indeed satisfy the Liebniz law on the set $W$ and in particular at $p\in W$. Showing that $D$ satisfies 1) and 3) is simpler.
As for your second question, let $W = U_{j_1}\cap U_{j_2}$. Again we can assume that $W$ makes intersection with only finitely many other $U_{j}$, call them $U_{j_3},\ldots U_{j_k}$. By simply writing out what $D_{V}X$ is on $W$:
\begin{equation}
D_{V}X = f_{j_1}D^{j_1}_{V}X + f_{j_2}D^{j_2}_{V}X+\ldots + f_{j_k}D^{j_k}_{V}X
\end{equation}
We see that it doesn't really depend on which coordinate neighbourhood we are considering (it involves information from all the coordinate neighbourhoods that intersect at $W$). 
Hope this helps
