Slight trivial change in notation: Let $P_n$ be the probability that every run of consecutive 1s in our string does not exceed $n$ plus the previous acumulated runs.
Then, the following linear recursion (equivalent to that of your second approach) holds:
$$P_n = \sum_{k=1}^n P_{n+k} \; 2^{-k} \hspace{12mm} n=1, 2 \ldots \tag{1}
$$
(this is obtained by summing over all the possible lengths of the first run; because, after the first run of length $k$, we have the equivant problem with $n$ replaced by $k+n$)
This recursion is rather tricky to solve, because our "initial value" would be $P_{\infty}=1$, and the recursion is not time-reversible. Here's my approach (perhaps there are more elegant procedures, I'm not really familiar with this).
I try a solution of the form:
$$P_n = 1 + a_1 2^{-n} + a_2 2^{-2n} + a_3 2^{-3n} + \cdots$$
Actually, we'll only keep the terms with exponents of the form $-(2^p-1)n$. That is, we set $a_i=0$ unless $i = 2^p-1$ for some integer $p$. Calling $Q(r, n) = 2^{-r \, n}$, we have:
$$P_n = Q(0,n) + a_1 \, Q(1 , n) + a_3 \, Q(3,n) + a_7 \, Q(7,n) + a_{15} \, Q(15,n) +\cdots$$
Now, it's straighfroward to check (geometric sum) that:
$$\sum_{k=1}^n Q(r,n+k) \; 2^{-k} = \frac{1}{2^{r+1}-1} \bigl( Q(r,n) - Q(2r+1,n) \bigr) $$
Hence, the recursion (1) induces the transformations $Q_0 \to Q_0 -Q_1$, $Q_1 \to \frac{1}{3}(Q_1 -Q_3)$, $Q_3 \to \frac{1}{7}(Q_3 -Q_7)$, etc, and thus the coefficients can be obtained:
$$\begin{align}
a_1 &= \frac{a_1}{3} - 1 \Rightarrow a_1 = -\frac{3}{2} \\
a_3 &= \frac{a_3}{7} - \frac{a_1}{3} \Rightarrow a_3 = \frac{1}{3} \frac{7}{6} \frac{3}{2}=\frac{7}{6 \times 2}\\
a_7 &= \frac{a_7}{15} - \frac{a_3}{7} \Rightarrow a_7 = - \frac{1}{7} \frac{15}{14} \frac{1}{3} \frac{7}{6} \frac{3}{2} =-\frac{15}{14 \times 6 \times 2} \\
a_{15} &= \cdots = \frac{31}{30 \times 14 \times 6 \times 2}\\
a_{2^p-1} &= \cdots = (-1)^p \frac{2^{p-1}-1}{(2^{p-1}-2) \times (2^{p-2}-2) \cdots \times 14 \times 6 \times 2}\\
\end{align}
$$
Or, more compact (perhaps not more illuminating) :
$$ a_{2^p-1} = \frac{2-2^{-p}}{(2;2)_p}$$
where $(a;q)_n$ is the q-Pochhammer_symbol.
Some numerical values:
$ a_1=-1.50$, $ a_3\approx0.58333$, $a_7\approx-0.08929 $, $a_{15}\approx 0.00615$, $a_{31}\approx-0.0002$
So, finally
$$ P_n = 1 - \frac{3}{2} 2^{-n} + \frac{7}{12} 2^{-3 n } + \cdots =\\
= 1 +\sum_{p=1}^\infty \frac{2-2^{-p}}{(2;2)_p}2^{-(2^p-1)n}$$
This converges quite quickly, specially for big $n$.
To get, according to the original question, the probability that some run is greater or equal than $m$, we'd evaluate
$$1 - P_{m-1} = 3 \, 2^{-m} - \frac{14}{3} 2^{-3 m} + \cdots $$
which in the first approximation coincides with the OP's estimate (using the expectation); it is quite a good approximation for, say, $n>4$.
n 1 2 3 4 5 6 7
p(n) exact 0.32222 0.63411 0.81364 0.90639 0.95314 0.97656 0.98828
p(n) approx 0.25000 0.62500 0.81250 0.90625 0.95312 0.97656 0.98828
Added Rigurous bounds for $P(n)$ can be obtained by noting that, with my definition, it's equivalent to
the probability that a sequence of iid geometric variables $x_i = 1, 2 \cdots$ attaining values in
$x_1 \le n$,$x_2 \le n + x_1 $,$x_3 \le n + x_1 + x_2$ ...
So
$$P(n) = \sum_{k_1}^{n} 2^{-k_1} \sum_{k_2}^{n+k_1} 2^{-k_2} \sum_{k_3}^{n+k_1+k_2} 2^{-k_3} \cdots$$
In particular, a lower bound for the $m$-th terms in this infinite nested product is obtained by setting the outer values in $k_1=1$, $k_2=1$..., so
$$P(n) \le \sum_{k_1}^{n} 2^{-k_1} \sum_{k_2}^{n+1} 2^{-k_2} \sum_{k_3}^{n+2} 2^{-k_3} \cdots = (1 - 2^{-n})(1 - 2^{-n-1})(1 - 2^{-n-2}) = \\ = \prod_{i=0}^\infty (1 - 2^{-n} 2^{-i})= S(2^{-n},2^{-1})
$$
where $S(a,q)=\prod_{i=0}^\infty (1 - a q^{i})$ is the q-Pochamer symbol or (q-series) QPochhammer[a,q]
We can check that this tends to 1 when $a\to 0$ by
using the bound $0<-\log(1-x)<2x$ for $0<x \le 1/2$, so $\log( S(2^{-n},2^{-1})$ is bounded between $0$ and $-2^{-n+1}$. Hence $\lim_{n\to\infty} P(n)=1$.
Added 2: Regarding uniqueness: We must prove that the recursion (1), with the boundary condition $P_{\infty}=1$ does not admit more than one solution (with $0 \le P_n \le 1)$. This is equivalent to prove that the recursion does not have a solution with $P_{\infty}=0$ except for the trivial one: $P_n=0$.
Now, suppose $P_{n_1}>0$. Then, the equation (1) implies that there exists at least one $n_2$ in $n_1 +1 \cdots 2 n_1$ with $P_{n_2} > P_{n_1}$. Applying the same to $n_2$, this implies that we can find an increasing subsequence of $P(n)$; hence its limit cannot be zero.
Best Answer
The solution manual is correct in stating that-
E(L1|X=0) = 1 + ($\frac{1}{p}$ - 1)
In this expression, the first 1 accounts for the "0" symbol at the very start of the sequence. The second term accounts for the number of zeroes until the first one is encountered. You are right, $\frac{1}{p}$ includes the final symbol "1" that we see in the sequence and we thus need to subtract one from $\frac{1}{p}$. However, you forgot to account for the first "0" symbol encountered at the very beginning.