3. There are $n$ ellipses centered at the origin with the coordinate axes as axes of symmetry, and the directrices of all ellipses are $x=1$. If the eccentricity of the $k$-th $(k=1,2, \cdots, n)$ ellipse is $e_{k}=2^{-k}$, then the sum of the major axes of these $n$ ellipses is $\qquad$ .