Let $n$ be a natural number, $$\begin{aligned}
& f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\
& f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } }
\end{aligned}$$ The regions between the functions defined in this form and the x-axis are given shaded in the figure below. Accordingly, what is the sum of the areas of all shaded regions in square units? A) $\frac { 2 } { 3 }$ B) $\frac { 3 } { 4 }$ C) $\frac { 5 } { 6 }$ D) $\frac { 8 } { 9 }$ E) $\frac { 11 } { 12 }$
Let $n$ be a natural number,
$$\begin{aligned}
& f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\
& f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } }
\end{aligned}$$
The regions between the functions defined in this form and the x-axis are given shaded in the figure below.
Accordingly, what is the sum of the areas of all shaded regions in square units?\\
A) $\frac { 2 } { 3 }$\\
B) $\frac { 3 } { 4 }$\\
C) $\frac { 5 } { 6 }$\\
D) $\frac { 8 } { 9 }$\\
E) $\frac { 11 } { 12 }$