Recursive sequence with binomial coefficients I have a sequence $\epsilon_i$ defined recursively for $i\ge 1$ as follows
\begin{eqnarray*}
\epsilon_1 &=& \frac{1}{p}\\
\epsilon_n &=& \frac{1}{1-(1-p)^n}\left( 1 + \sum_{j=1}^{n-1} \binom{n}{j}(1-p)^{j} p^{n-j}\epsilon_{j}\right)
\end{eqnarray*}
with $p \in (0,1)$. I want to prove that $\epsilon_i$ asymptotically grows as $\log i$, but all my attempts failed. I am unable to get a closed form for the sequence. I know from plotting $\epsilon_i$ for values 1 to 50 that my claim is true. Any hint on how to solve this problem is welcome. Actually I only need the value for $p=0.5$.
This sequence came up when I computed the expected value for a random variable.
 A: Update: For fixed $n$, the function $x\mapsto 1-(1-q^x)^n$ is decreasing in $x>0$, and 
therefore 
$$\epsilon_n\geq \int^\infty_0  1-(1-q^x)^n\,dx\geq \epsilon_n-1.$$
Now, with a change of variables $x=w \log(n)$, the integral becomes
$$\int^\infty_0  1-(1-q^x)^n\,dx=\log(n) \int_0^\infty 1-\left(1-{1\over n^{w \log(1/q)}}\right)^n\,dw.$$
The integrand converges to 1 if $w < 1/\log(1/q)$ and zero if $w > 1/\log(1/q)$.
By dominated convergence
$$\int_0^\infty 1-\left(1-{1\over n^{w \log(1/q)}}\right)^n\,dw\to1/\log(1/q),$$
and we deduce that asymptotically 
$$\epsilon_n\approx \log(n)/\log(1/q).$$ 

Start with $n$ independent coins with probability $p$ of showing heads.
Toss them all, and put aside  those that show "heads". Retoss the remaining 
coins, and repeat until all coins show "heads". 
Your $\epsilon_n$ is the expected number of trials in this experiment.
Here is an explicit formula:
$$\epsilon_n=\sum_{j\geq 1}{n\choose j}(1-q^j)^{-1}(-1)^{(j+1)}=\sum_{k\geq 0}[1-(1-q^k)^n], $$
where $q=1-p$. Perhaps the asymptotics can be derived from this expression.
In the reference given below the author says "A graph of the data in TABLE 2 
strongly suggests that E[Y] (i.e., your $\epsilon_n$) is a logarithmic function of $n$." 
But no proof is offered.
Tossing Coins Until All Are Heads
by  John Kinney in 
Mathematics Magazine, Vol. 51, No. 3 (May, 1978), pp. 184-186.
