= Binomial branching survival correction
{title2=$\rho=2\varepsilon+O(\varepsilon^2+\varepsilon/n)$}
For $p=(1+\varepsilon)/n$, the <branching survival probability> satisfies $1-\rho=(1-p\rho)^n$. If $(1+\varepsilon)^2<n$, comparison with a <Poisson branching process> and expansion of the <logarithm> give
$$
\frac{2\varepsilon}{1+2\varepsilon}\leq\rho\leq\frac{2\varepsilon}{1-(1+\varepsilon)^2/n}.
$$
For the upper bound use $\varepsilon=\sum_{j\geq1}(1-np^{j+1})\rho^j/(j+1)$ and retain its first nonnegative term. For fixed $n>1$, the leading term is instead $2n\varepsilon/(n-1)$. In particular $n=2$ gives $\rho=4\varepsilon/(1+\varepsilon)^2$, showing that the uncorrected exact bound $\rho\leq2\varepsilon$ is false for small positive $\varepsilon$.
For $0<\varepsilon\leq1/4$ and $n\varepsilon\geq2$, the uncorrected exact upper bound does hold for the <binomial branching process>. Here $np^2\leq\varepsilon$ and $np^3\leq1/4$, so at $x=2\varepsilon$ the first two terms of the nonnegative series give $\varepsilon(1-np^2)+(4/3)\varepsilon^2(1-np^3)\geq\varepsilon$. Monotonicity of that series yields $\rho\leq2\varepsilon$. This recovers the intended large-$n$ bracket without claiming it for every finite reproduction law.
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