Is there a closed form for the distribution of $T = \sum_{k=0}^\infty 2^{-k} X_k$? Let
$$T = \sum_{k=0}^\infty 2^{-k} X_k$$
Where each $X_k \sim \text{Exp}(1)$ is i.i.d. as an exponential random variable with $\lambda = 1$. Is there a closed form for the pdf or cdf of $T$?
 A: A few notes that might be helpful.
Suppose $0 < a_0 < a_1 < \cdots$ and that
$X$ has density
\begin{equation*}
 f(x) = \sum_{m=0}^{M-1} C_m e^{-a_m x}
\end{equation*}
on $x \geq 0$, where the $C_m$ are not necessarily nonnegative, and that $Y$ is
independent and distributed as $a_M e^{-a_M y}$, $y \geq 0$.
Then $Z = X + Y$ has density
\begin{align*}
 f_Z(z) &= \int_0^z f(x) a_M e^{-a_M(z - x)}\, dx \\
 &= \int_0^z\, \sum_{m=0}^{M-1}a_M C_m e^{-a_mx}e^{-a_Mz + a_Mx}\, dx \\
 &= \sum_{m=0}^{M-1}a_M C_m e^{-a_Mz}\int_0^z\, e^{-a_mx}e^{a_Mx}\, dx \\
 &= \sum_{m=0}^{M-1}a_M C_m e^{-a_Mz}\int_0^z\, e^{(a_M - a_m)x}\, dx \\
 &= \sum_{m=0}^{M-1}\frac{a_M C_m}{a_M - a_m} e^{-a_Mz} (e^{(a_M - a_m)z} - 1) \\
 &= \sum_{m=0}^{M-1}\frac{a_M C_m}{a_M - a_m} (e^{-a_mz} - e^{-a_Mz}) \\
 &= \sum_{m=0}^{M-1}\frac{a_M C_m}{a_M - a_m} e^{-a_mz} 
 - \bigl( \sum_{m=0}^{M-1}\frac{a_M C_m}{a_M - a_m}\bigr) e^{-a_Mz}
\end{align*}
which shows $f_Z$ has a form similar to the above:
\begin{equation*}
 f_Z(z) = \sum_{m=0}^M C'_m e^{-a_m z}
\end{equation*}
with
\begin{equation*}
 C'_m = \frac{a_M C_m}{a_M - a_m}
\end{equation*}
for $m = 0, \dots, M-1$ and
\begin{equation*}
 C'_M = -\sum_{m=0}^{M-1} \frac{a_M C_m}{a_M - a_m}.
\end{equation*}
This lends itself to an iteration constructing the limiting density as $M \rightarrow \infty$, at least when it exists.
The original question involves $a_m = 2^m$ so we expect a density of the form
\begin{equation*}
 f_\infty(z) = D_0e^{-z} + D_1e^{-2z} + D_2e^{-4z} + D_3e^{-8z} + \dots
\end{equation*}
The recursion for the leading coefficient yields
$D_0 = 1/ \prod_{k=1}^\infty (1 - 2^{-k}) \approx 3.462746619$.

ADDED
The first few coefficients, computed using the recurrence above, are
\begin{align*}
m &\qquad\qquad D_m \\
0 &\qquad +3.46274662e+00 \\
1 &\qquad -6.92549324e+00 \\
2 &\qquad +4.61699549e+00 \\
3 &\qquad -1.31914157e+00 \\
4 &\qquad +1.75885543e-01 \\
5 &\qquad -1.13474544e-02 \\
6 &\qquad +3.60236646e-04 \\
7 &\qquad -5.67301805e-06 \\
8 &\qquad +4.44942599e-08 \\
9 &\qquad -1.74147636e-10
\end{align*}
