Survival Probability of a Population A population starts with one amoeba. In each generation, each amoeba divides in two with probability $\frac{1}{2}$, or dies, with probability $\frac{1}{2}$. Let $p_n$ be the probability that the population will survive $n$ generations.
$(a)$ What is $p_4$?
$(b)$ Find the limit $a = \lim_{n\to\infty} np_n$
$(c)$ Show that $p_n = \frac{a}{n} + \frac{b \space \log(n)}{n^2} + \mathcal{O} \left(\frac{1}{n^2}\right) $ as $n \to \infty$, and find $b$.
I have had a hard time trying to find a method to solve the problem so any help is much appreciated! 
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Each amoeba can generate $0$ amoebas or $2$ amoebas with the same probability
$\pars{~{1 \over 2}~}$. Then, the probability $P_{n}$ that an amoeba generate
$n$ amoebas in the next generation is given by
$$
P_{n} = {1 \over 2}\pars{\delta_{n,0} + \delta_{n,2}}
$$
The probability $P_{N \to N'}$ that $N$ amoebas generate $N'$ amoebas in the next generation is given by:
\begin{align}
P_{N \to N'}
&=
\left.\sum_{n_{1} = 0}^{\infty}P_{n_{1}}\ldots\sum_{n_{N} = 0}^{\infty}P_{n_{N}}
\right\vert_{\sum_{i = 1}^{N}n_{i} = N'}
\\[3mm]&=
\sum_{n_{1} = 0}^{\infty}P_{n_{1}}\ldots\!\!\!\sum_{n_{N} = 0}^{\infty}P_{n_{N}}
\int_{\verts{z} = 1}{1 \over z^{\pars{\sum_{i = 1}^{N}n_{i}\ -\ N'\ +\ 1}}}
\,{\dd z \over 2\pi\ic}
\\[3mm]&=
\int_{\verts{z} = 1}{\dd z \over 2\pi\ic}\,{1 \over z^{1 - N'}}
\pars{\sum_{n = 0}^{\infty}P_{n}\,{1 \over z^{n}}}^{N}
=
\int_{\verts{z} = 1}{\dd z \over 2\pi\ic}\,{1 \over z^{1 - N'}}
\pars{{1 \over 2}\,{1 \over z^{0}} + {1 \over 2}\,{1 \over z^{2}}}^{N}
\\[3mm]&=
{1 \over 2^{N}}\int_{\verts{z} = 1}{\dd z \over 2\pi\ic}\,
{1 \over z^{1 - N' + 2N}}
\pars{1 + z^{2}}^{N}
=
{1 \over 2^{N}}\int_{\verts{z} = 1}{\dd z \over 2\pi\ic}\,
{1 \over z^{1 - N' + 2N}}
\sum_{n = 0}^{N}{N \choose n}z^{2n}
\\[3mm]&=
{1 \over 2^{N}}\sum_{n = 0}^{N}{N \choose n}\int_{\verts{z} = 1}{\dd z \over 2\pi\ic}\,
{1 \over z^{1 - N' + 2N - 2n}}
=
{1 \over 2^{N}}\sum_{n = 0}^{N}{N \choose n}
\delta_{2n,2N - N'}
\end{align}
The last sum is non zero whenever
$$
0 \leq 2N - N' \leq 2N\quad\mbox{and}\quad N'\ \mbox{even}
$$
Then
$$
P_{N \to N'}
=
\left\lbrace%
\begin{array}{lcl}
{1 \over 2^{N}}{N \choose \vphantom{\large A^{A}}{N' \over 2}}
& \mbox{if} &
\left\lbrace%
\begin{array}{l}
0 \leq N' \leq 2N
\\
N'\ \mbox{even}
\end{array}\right.
\\[2mm]
0 && \mbox{otherwise}
\end{array}\right.
$$
Once we know $P_{N \to N'}$ we can analyze several situations.
