In this question you may use the results from II.B.2d. a) Let $\alpha$ be in $]0, 1/2[$. For $n$ in $\mathbb{N}^{*}$, we set $$A_{n} = \left\{k \in \{0, \ldots, n\}, \left|k - \frac{n}{2}\right| \geqslant \alpha n\right\}, \quad U_{n} = \sum_{k \in A_{n}} \binom{n}{k}$$ Determine the limit of the sequence $\left(U_{n}^{1/n}\right)_{n \geqslant 1}$. b) Let $\lambda$ be in $\mathbb{R}^{+*}$, $\alpha$ be in $]\lambda, +\infty[$. For $n$ in $\mathbb{N}^{*}$, we set $$T_{n} = \sum_{\substack{k \in \mathbb{N} \\ k \geqslant \alpha n}} \frac{n^{k} \lambda^{k}}{k!}$$ Determine the limit of the sequence $\left(T_{n}^{1/n}\right)_{n \geqslant 1}$.
In this question you may use the results from II.B.2d.
a) Let $\alpha$ be in $]0, 1/2[$. For $n$ in $\mathbb{N}^{*}$, we set
$$A_{n} = \left\{k \in \{0, \ldots, n\}, \left|k - \frac{n}{2}\right| \geqslant \alpha n\right\}, \quad U_{n} = \sum_{k \in A_{n}} \binom{n}{k}$$
Determine the limit of the sequence $\left(U_{n}^{1/n}\right)_{n \geqslant 1}$.
b) Let $\lambda$ be in $\mathbb{R}^{+*}$, $\alpha$ be in $]\lambda, +\infty[$. For $n$ in $\mathbb{N}^{*}$, we set
$$T_{n} = \sum_{\substack{k \in \mathbb{N} \\ k \geqslant \alpha n}} \frac{n^{k} \lambda^{k}}{k!}$$
Determine the limit of the sequence $\left(T_{n}^{1/n}\right)_{n \geqslant 1}$.