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 $\qquad$ .
3. $2-2^{1-n}$.

Let the semi-major axis of the $k$-th ellipse be $a_{k}$, and the semi-focal distance be $c_{k}$. According to the problem,
$$
\begin{array}{l}
\frac{a_{k}^{2}}{c_{k}}=1, e_{k}=\frac{c_{k}}{a_{k}}=2^{-k} \\
\Rightarrow a_{k}=2^{-k} \\
\Rightarrow \sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n} 2^{-k}=1-2^{-n} .
\end{array}
$$

Therefore, the sum of the major axes of these $n$ ellipses is $2-2^{1-n}$.