The following GeoGebra applet uses $n=2^k$.

Martin Flashman, Jan. 21, 2014. Created with GeoGebra

In the GeoGebra application, $e \approx (1+ \frac 1 {2^k})^{2^k}
$. Notice how the arrows on the mapping diagrams are paired with
the points on the graph.

You can use the slider to move the point for $k$ on the mapping
diagram to see how the function value for the estimate for $e$
changes both on the diagram and on the graph.