We consider the sequence $\left(v_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, v_{n}=\left(1+\frac{1}{2^{n}}\right)^{2^{n}}$.
a) Show that in the neighbourhood of $+\infty, v_{n}=\mathrm{e}-\frac{\mathrm{e}}{2^{n+1}}+o\left(\frac{1}{2^{n}}\right)$.
b) Show that the sequence $(v_{n})$ belongs to $E^{c}$ and give its convergence rate.