What's the $\frac{1}{2}\bigg(e+\frac{1}{e}\bigg)$ I encountered in context of martingales of Bernoulli r.v.s.?
Such as:
$\epsilon_1, \epsilon_2, ...$
i.d. Bernoulli r.v.s.
Let $M_0=1$ and $M_n = e^{\epsilon_1+...+\epsilon_n-cn}$ s.t. $\mathbb{E}(e^{\epsilon_1-c})=1$ with $c=\log\bigg( \frac{e+\frac{1}{e}}{2}\bigg)$.
I.e. where does this quantity come from?